Research ArticleFuzzy Chromatic Polynomial of Fuzzy Graphs with Crisp andFuzzy Vertices Using 120572-Cuts

Mamo Abebe Ashebo and V N Srinivasa Rao Repalle

Department of Mathematics Wollega University Nekemte Ethiopia

Correspondence should be addressed to Mamo Abebe Ashebo mamoabebe37gmailcom

Received 24 January 2019 Revised 28 March 2019 Accepted 2 April 2019 Published 2 May 2019

Academic Editor Antonin Dvorak

Copyright copy 2019 Mamo Abebe Ashebo and V N Srinivasa Rao Repalle This is an open access article distributed under theCreative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium providedthe original work is properly cited

Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system examscheduling register allocation etc In this paper the concept of fuzzy chromatic polynomial of fuzzy graph is introduced anddefined based on 120572-cuts of fuzzy graph Two different types of fuzziness to fuzzy graph are considered in the paper The first typewas fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edgeset Depending on this the fuzzy chromatic polynomials for some fuzzy graphs are discussed Some interesting remarks on fuzzychromatic polynomial of fuzzy graphs have been derived Further some results related to the concept are proved Lastly fuzzychromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained

1 Introduction

Nowadays many real world problems cannot be properlymodeled by a crisp graph theory since the problems containuncertain information The fuzzy set theory anticipated byZadeh [1] is used to handle the phenomena of uncertaintyin real life situation A lot of works have been done in fuzzyshortest path problems using type 1 fuzzy set in [2ndash5] Deyet al [6] introduced interval type 2 fuzzy set in the fuzzyshortest path problems Recently in [7] the authors proposeda genetic algorithm for solving fuzzy shortest path problemwith interval type 2 fuzzy arc lengths Some researchersalso used the fuzzy set theory to touch the uncertainty incrisp graphs Kaufmann [8] proposed the first definition offuzzy graph in 1973 based on Zadehrsquos fuzzy relations LaterRosenfeld [9] introduced another elaborated definition offuzzy graph with fuzzy vertex set and fuzzy edge set in1975 He developed the theory of fuzzy graph After thatBhattacharya [10] has established some connectivity conceptsregarding fuzzy cut nodes and fuzzy bridges Bhutani [11]has studied automorphisms on fuzzy graphs and certainproperties of complete fuzzy graphs Also Mordeson andNair [12] introduced cycles and cocycles of fuzzy graphsSeveral authors including Sunitha and Vijayakumar [13 14]

Bhutani and Rosenfeld [15]Mathew and Sunitha [16] Akram[17] and Akram and Dudek [18] have introduced numerousconcepts in fuzzy graphs Fuzzy graph theory has severalapplications in various fields like clustering analysis databasetheory network analysis information theory etc [19]

Coloring of fuzzy graphs plays a vital role in theoryand practical applications It is mainly studied in combina-torial optimization problems like traffic light control examscheduling register allocation etc [20] Fuzzy coloring of afuzzy graph was defined by authors Eslahchi and Onagh in2004 and later developed by them as fuzzy vertex coloring[21] in 2006 Another approach of coloring of fuzzy graphswas introduced byMunoz et al [22]The authors have definedthe chromatic number of a fuzzy graph Incorporating theabove two approaches of coloring of fuzzy graph Kishoreand Sunitha [23] introduced chromatic number of fuzzygraphs and developed algorithm Dey and Pal [24] intro-duced the vertex coloring of a fuzzy graph using 120572-cutsIn [25] they have used the vertex coloring of fuzzy graphto classify the accidental zone of a traffic control Furtherin [26] the authors proposed genetic algorithm to findthe robust solutions for fuzzy robust coloring problem Theauthors Ananthanarayanan and Lavanya [20] introducedfuzzy chromatic number and fuzzy total chromatic number

HindawiAdvances in Fuzzy SystemsVolume 2019 Article ID 5213020 11 pageshttpsdoiorg10115520195213020

2 Advances in Fuzzy Systems

of a fuzzy graph using 120572-cuts Rosyida et al [27] proposeda new approach to determine fuzzy chromatic number offuzzy graph through its 120575-chromatic number Samanta et al[28] also introduced a new concept for coloring of fuzzygraphs using fuzzy color Lately Dey et al [29] introduced theconcept of vertex and edge coloring of vague graphswhich arethe generalized structure of fuzzy graphs

In the literature to the best of our knowledge thereis no study on the fuzzy chromatic polynomial of fuzzygraphs Therefore in this paper we consider the chromaticpolynomial in fuzzy graph called fuzzy chromatic polyno-mial of fuzzy graph Based on this we define the conceptof fuzzy chromatic polynomial of a fuzzy graph using 120572-cuts of fuzzy graph Also we determine the fuzzy chromaticpolynomial for fuzzy graphs with crisp and fuzzy vertex setTo determine the fuzzy chromatic polynomial the classicalmethod for computing the chromatic polynomial of crispgraph is used Next we derive some interesting remarks onfuzzy chromatic polynomial of fuzzy graph with crisp andfuzzy vertices Further we provemore elegant results on fuzzychromatic polynomial of fuzzy graphs Finally we study afuzzy chromatic polynomial for complete fuzzy graphs andfuzzy cycles

The rest of paper is organized as follows In Section 2some basic definitions and elementary concepts of fuzzyset fuzzy graph and coloring of fuzzy graphs are reviewedIn Section 3 fuzzy chromatic polynomial of a fuzzy graphusing 120572-cut of fuzzy graph is defined Also fuzzy chromaticpolynomials for fuzzy graphs with crisp and fuzzy verticesare determined In Section 4 more results on fuzzy chro-matic polynomials are proved Furthermore fuzzy chromaticpolynomial for complete fuzzy graphs and fuzzy cycles arestudied Finally the paper is concluded in Section 5

2 Preliminaries

In this section some basic aspects that are necessary for thispaper are included These preliminaries are given in threesubsections

21 Basic Definitions and Concepts on Vertex Coloring andChromatic Polynomial In this subsection some basic def-initions and concepts of vertex coloring and chromaticpolynomials are reviewed [30 31]

Definition 1 Let G = (VE) be a graph A vertex-coloringof G is an assignment of a color to each of the vertices ofG in such a way that adjacent vertices are assigned differentcolors If the colors are chosen from a set of 119896 colors then thevertex-coloring is called a 119896-vertex-coloring abbreviated to119896-coloring whether or not all 119896 colors are used

Definition 2 If G has a 119896-coloring then 119866 is said to be 119896-colorable

Definition 3 The smallest 119896 such that G is 119896-colorable iscalled the chromatic number of G denoted by 120594(G)

Definition 4 Let G be a simple graph The chromatic poly-nomial of G is the number of ways we can achieve a propercoloring on the vertices of G with the given 119896 colors and it isdenoted by 119875(G 119896) It is a monic polynomial in 119896with integercoefficients whose degree is the number of vertices of G

If we are given an arbitrary simple graph it is usuallydifficult to obtain its chromatic polynomial by examining thestructure of a graph (by inspection) The following theoremgives us a systematic method for obtaining the chromaticpolynomial of a simple graph in terms of the chromaticpolynomial of null graphs

Theorem 5 Let G be a simple graph and let G minus e and Gebe the graphs obtained from G by deleting and contracting anedge e en

P (G k) = P (G minus e k) minus P (Ge k) (1)

22 Basic Definitions on Fuzzy Set and Fuzzy Graphs In thissubsection some basic definitions on fuzzy set and fuzzygraphs are reviewed [1 9 19 32]

Definition 6 A fuzzy set 119860 defined on a nonempty set 119883 isthe family119860 = (119909 120583119860(119909)) | 119909 isin 119883 where 120583119860 119883 997888rarr 119868 is themembership function In classical fuzzy set theory the set I isusually defined as the interval [0 1] such that

120583119860 (119909) =

1 119909 isin 119860

0 119909 notin 119860(2)

It takes any intermediate value between 0 and 1 represents thedegree in which 119909 isin 119860 The set 119868 could be discrete set of theform 119868 = 0 1 119896 where 120583119860(119909) lt 120583119860(1199091015840) indicates thatthe degree of membership of x to A is lower than the degreeof membership of 1199091015840

Definition 7 120572-cut set of fuzzy set 119860 is defined as 119860120572 is madeup of members whose membership is not less than 120572 119860120572 =119909 isin 119883 | 120583119860(119909) ge 120572 120572-cut set of fuzzy set is crisp set

Definition 8 A fuzzy graph 119866 = (119881 120590 120583) is an algebraicstructure of nonempty set119881 together with a pair of functions120590 119881 997888rarr [0 1] and 120583 119881times119881 997888rarr [0 1] such that for all 119906 V isin119881 120583(119906 V) le 120590(119906) and 120590(V) and 120583 is a symmetric fuzzy relationon 120590 Here 120590(119906) and 120583(119906 V) represent the membership valuesof the vertex 119906 and of the edge (119906 V) in 119866 respectively

In this paper we denote 119906 and V = min119906 V and 119906 orV = max119906 V Here we considered fuzzy graph 119866 is simple(with no loops and parallel edges) finite and undirected 120583 isreflective (that is 120583(V V) = 120590(V) for all V isin 119881) and symmetric(that is 120583(119906 V) = 120583(V 119906) for all (119906 V) isin 119864)

Note that a fuzzy graph is a generalization of crisp graphin which 120590(V) = 1 for all V isin 119881 and 120583(119906 V) = 1 if (119906 V) isin 119864and 120583(119906 V) = 0 if (119906 V) notin 119864 So all the crisp graphs are fuzzygraphs but all fuzzy graphs are not crisp graphs

Advances in Fuzzy Systems 3

Definition 9 The fuzzy graph1198661015840 = (119881 1205901015840 1205831015840) is called a fuzzysubgraph of G if for each two elements u V isin 119881 we have1205901015840(119906) le 120590(119906) and 1205831015840(119906 V) le 120583(119906 V)

Definition 10 The fuzzy graph 119866 = (119881 120590 120583) is calledconnected if for every two elements u V isin 119881 there exists asequence of elements 1199060 1199061 119906119898 such that 1199060 = 119906 119906119898 = Vand 120583(119906119894 119906119894+1) gt 0 (0 le 119894 le 119898 minus 1)

Definition 11 For any fuzzy graph 119866 = (119881 120590 120583) let 119871 =120590(119906) gt 0 | 119906 isin 119881 cup 120583(119906 V) gt 0 | 119906 = V 119906 V isin 119881with 119896 elements Now assume 119871 = 1205721 1205722 120572119896 such that1205721 lt 1205722 lt sdot sdot sdot lt 120572119896 The sequence 1205721 1205722 120572119896 and the set119871 are called the fundamental sequence and the fundamentalset (or level set) of 119866 respectively

Definition 12 The underlying crisp graph 119866lowast = (119881lowast 119864lowast) of afuzzy graph119866 = (119881 120590 120583) is such that119881lowast = 119906 isin 119881 | 120590(119906) gt 0and 119864lowast = (119906 V) | 120583(119906 V) gt 0

Definition 13 For 120572 isin [0 1] 120572-cut graph of a fuzzy graph119866 =(119881 120590 120583) is a crisp graph119866120572 = (119881120572 119864120572) such that119881120572 = 119906 isin 119881 |120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 It is obvious that afuzzy graph will have a finite number of different 120572-cuts

23 Basic Definitions in Fuzzy Coloring of Fuzzy Graphs Theconcept of chromatic number of fuzzy graph was introducedby Munoz et al [22] The authors considered fuzzy graphswith crisp vertex set that is fuzzy graphs for which 120590(V) = 1for all V isin 119881 and edges with membership degree in [0 1]

Definition 14 (see [22]) If 119866 = (119881 120583) is such a fuzzy graphwhere 119881 = 1 2 3119899 and 120583 is a fuzzy number on the set ofall subsets of 119881 times 119881 assume 119868 = 119860 cup 0 where 119860 = 1205721 lt1205722 lt lt 120572119896 is the fundamental set (level set) of 119866 Foreach 120572 isin 119868 119866120572 denote the crisp graph G120572 = (119881 119864120572) where119864120572 = 119894119895|1 le 119894 lt 119895 le 119899 120583(119894 119895) ge 120572 and 120594120572 = 120594(119866120572) denotethe chromatic number of crisp graph 119866120572

By this definition the chromatic number of fuzzy graphsG is the fuzzy number 120594(119866) = (119894 V(119894))119894 isin 119883 where V(119894) =max120572 isin 119868 | 119894 isin 119860120572 and 119860120572 = 1 2 3 120594120572

Later Eslahchi and Onagh [21] introduced fuzzy vertexcoloring of fuzzy graph They defined fuzzy chromatic num-ber as the least value of 119896 for which the fuzzy graph 119866 has119896-fuzzy coloring as follows

Definition 15 (see [21]) A family Γ = 1205741 1205742 120574119896 of fuzzysets on a set 119881 is called a 119896-fuzzy coloring of 119866 = (119881 120590 120583) if

(i) orΓ = 120590

(ii) 120574119894 and 120574119895 = 0

(iii) for every strong edge (119909 119910) (ie 120583(119909 119910) gt 0 ) of 119866min120574119894(119909) 120574119894(119910) = 0 (1 le 119894 le 119896)

Definition 16 (see [21]) The minimum number 119896 for whichthere exists a 119896-fuzzy coloring is called the fuzzy chromaticnumber of 119866 denoted as 120594119891(119866)

Incorporating the features of the above two definitionsKishore and Sunitha [23] modified the chromatic number120594(119866) of fuzzy graph as follows

Definition 17 (see [23]) For each 120572120598119868 119866120572 denote the crispgraph 119866120572 = (120590120572 120583120572) and 120594120572 = 120594(119866120572) denote the chromaticnumber of crisp graph 119866120572 The chromatic number of fuzzygraph 119866 = (119881 120590 120583) is the number 120594(119866) = max120594(119866120572) | 120572 isin119860

3 Fuzzy Chromatic Polynomial ofa Fuzzy Graph

A fuzzy chromatic polynomial is a polynomial which isassociated with the fuzzy coloring of fuzzy graphs There-fore chromatic polynomial in fuzzy graph is called fuzzychromatic polynomial of fuzzy graph In this section wedefine the concept of fuzzy chromatic polynomial of afuzzy graph based on 120572-cuts of fuzzy graph which are crispgraphs Furthermore we determine the fuzzy chromaticpolynomials for some fuzzy graphs with crisp and fuzzyvertices

Crisp vertex coloring and the chromatic polynomial of 120572-cut graph of the fuzzy graph are defined as follows

Definition 18 Let 119866 = (119881 120590 120583) be a fuzzy graph and 119866120572denote120572-cut graph of the fuzzy graph119866which is a crisp graph119866120572 = (119881120572 119864120572) 120572 isin [0 1]A function119862119896120572 119881120572 997888rarr 1 2 119896119896 isin N is called a 119896-coloring (crisp vertex coloring) of 119866120572 if119862119896120572(V119894) = 119862119896120572(V119895) whenever the vertices V119894 and V119895 are adjacentin 119866120572

Definition 19 The number of distinct k-coloring on thevertices of 119866120572 is called the chromatic polynomial of 119866120572 It isdenoted by 119875(119866120572 119896)

Let 119866120572 | 120572 isin 119868 be the family of 120572-cuts sets of 119866where the 120572-cut of a fuzzy graph is the crisp graph119866120572Henceany crisp k-coloring 119862119896120572 can be defined on the vertex set of119866120572 The k-coloring function of the fuzzy graph 119866 is definedthrough this sequence For each 120572 isin 119868 let the fuzzy chromaticpolynomial of 119866 be defined through a monotone family ofsets

Fuzzy chromatic polynomial of a fuzzy graph is definedas follows

Definition 20 Let 119866 be a fuzzy graph The fuzzy chromaticpolynomial of119866 is defined as the chromatic polynomial of itscrisp graphs 119866120572 for 120572 isin 119868 It is denoted by 119875119891120572 (119866 119896)

That is 119875119891120572 (119866 119896) = 119875(119866120572 119896) forall120572 isin 119868

31 Fuzzy Chromatic Polynomial of Fuzzy Graph with CrispVertices In this subsection we present fuzzy chromaticpolynomial of fuzzy graphs with crisp vertices and fuzzyedges

A fuzzy graph 119866 with crisp vertices and fuzzy edges and120572-cut graph of 119866 are defined as follows

4 Advances in Fuzzy Systems

Definition 21 A fuzzy graph is defined as a pair 119866 = (119881 120583)such that

(1) V is the crisp set of vertices (that is120590(119906) = 1 forall119906 isin 119881)(2) the function 120583 119881 x 119881 997888rarr [0 1] is defined by

120583(119906 V) le 120590(119906) and 120590(V) for all 119906 V isin 119881

Definition 22 Let 119866 = (119881 120583) be a fuzzy graph For 120572 isin 119868 120572-cut graph of the fuzzy graph 119866 is defined as the crisp graph119866120572 = (119881 119864120572) where 119864120572 = (119906 V) 119906 V isin 119881 | 120583(119906 V) ge 120572

Example 23 Consider the fuzzy graph 119866 with crisp verticesand fuzzy edges in Figure 1

In 119866 we consider 119868 = 0 02 03 05 08 1 for each120572 isin 119868 we have a crisp graph119866120572 and its chromatic polynomialwhich is the fuzzy chromatic polynomial of the fuzzy graph119866is obtained (see Figure 2) (The integers in the brackets denotethe number of ways of coloring the vertices)

Remark 24 The fuzzy chromatic polynomial depends on thevalues of 120572 which means the fuzzy chromatic polynomialvaries for the same fuzzy graph 119866 for different values of 120572

For the fuzzy graph119866 in Example 23 the fuzzy chromaticpolynomial varies for different values of 120572 as shown below

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0

119896 (119896 minus 1) (119896 minus 2)2 120572 = 02

119896 (119896 minus 1)2 (119896 minus 2) 120572 = 03

119896 (119896 minus 1)3 120572 = 05

1198962 (119896 minus 1)2 120572 = 08

1198964 120572 = 1

(3)

Remark 25 The fuzzy chromatic polynomial of a fuzzy graphdoes not need to be decreased when the value of 120572 increasesLet us consider 1205721 = 02 1205722 = 03 1205723 = 05 1205724 = 08 and1205725 = 1 for the fuzzy graph 119866 in Example 23 but 119875119891120572119894(119866 119896)119896 ge 120594120572119894 where 120594120572119894 is the chromatic number of the crispgraph 119866120572119894 i=12345 is not decreased as 120572119894 increases Forinstance 1198751198911205721(119866 1205941205721) = 6 1198751198911205722(119866 1205941205722) = 12 1198751198911205723(119866 1205941205723) = 21198751198911205724(119866 1205941205724) = 4 and 1198751198911205725(119866 1205941205725) = 1

Remark 26 Let 119866 be a fuzzy graph with n vertices Then for119866 with crisp vertices and fuzzy edges and for 120572 = 1 119875119891120572 (119866 119896)is not always 119875(119873119899 119896) where119873119899 is the null crisp graph withn vertices which is clear from the following example

Example 27 Consider the fuzzy graph119866 given in Figure 3(a)In Figure 3(a) for 120572 = 1 119866120572 is a crisp graph in Figure 3(b)But 119866120572 is not a null crisp graph with three vertices Thus119875119891120572 (119866 119896) = 119875(1198733 119896)

The following result gives the degree of fuzzy chromaticpolynomial of fuzzy graph with crisp vertices and fuzzy edgesare equal to the number of vertices in crisp vertex set

02

030508

08 Ｐ1

Ｐ2Ｐ3

Ｐ4

Figure 1 The fuzzy graph 119866 with crisp vertices and fuzzy edges

Lemma 28 Let119866 = (119881 120583) be a fuzzy graph with crisp verticesand fuzzy edges en the degree of 119875119891120572 (119866 119896) = |119881| for all 120572 isin119868

Proof Let 119866 = (119881 120583) be a fuzzy graph with crisp verticesand fuzzy edges Now 119866120572 = (119881 119864120572) where 119864120572 = (119906 V) |120583(119906 V) ge 120572 We know that the degree of the chromaticpolynomial of a crisp graph 119866120572 is the number of vertices of119866120572 That is the degree of 119875(119866120572 119896) = |119881| By Definition 20we have 119875119891120572 (119866 119896) = 119875(119866120572 119896) for 120572 isin 119868 From this it followsthat the degree of 119875119891120572 (119866 119896) is equal to the degree of 119875(119866120572 119896)Therefore 119875119891120572 (119866 119896) = |119881|

32 Fuzzy Chromatic Polynomial of Fuzzy Graph with FuzzyVertices In Section 31 we introduced the fuzzy chromaticpolynomial of fuzzy graph 119866 = (119881 120583) where 119881 is crispset vertices and 119864 is fuzzy set of edges In this subsectionwe present fuzzy chromatic polynomial of fuzzy graphs withfuzzy vertex set and fuzzy edge set Here we use Definition 8for fuzzy graph and Definition 13 for 120572-cut graph which arepresented in Section 2

Example 29 Consider a fuzzy graph 119866 with fuzzy vertex setand fuzzy edge set in Figure 4

In 119866 we consider 119868 = 0 02 03 04 06 08 for each120572 isin 119868 we have a crisp graph119866120572 and its chromatic polynomialwhich is the fuzzy chromatic polynomial of the fuzzy graph119866is obtained (see Figure 5) (The integers in the brackets denotethe number of ways of coloring the vertices)

Remark 30 The fuzzy chromatic polynomial varies for thesame fuzzy graph 119866 for different values of 120572 It is clear fromExample 29 Here the fuzzy chromatic polynomial of thefuzzy graph in Example 29 is given below

119875119891120572 (119866 119896)

=

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) (119896 minus 4) 120572 = 0

119896 (119896 minus 1) (119896 minus 2)3 120572 = 02

1198962 (119896 minus 1)3 120572 = 03

1198965 120572 = 04

1198963 120572 = 06

119896 120572 = 08

(4)

Advances in Fuzzy Systems 5

(k )

(k-1)(k-2)

(k-1) (k-1)

(k-1)

(k-1)

( k )

(k )

(k-1)(k-2)

(k-2)(k )

(k-1)(k-2)

(k-3)

(k )

(k )

(k )

(k )(k-1) (k-1)

(k )(k )

Ｐ1

Ｐ2

Ｐ2

Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4 Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4Ｐ1

Ｐ3

Ｐ4

= 1 Pf (G k) = k4

= 0 Pf (G k) = k(k minus 1)(k minus 2)(k minus 3)

= 03 Pf (G k) = k (k minus 1)2 (k minus 2)

= 08 Pf (G k) = k2 (k minus 1)2

= 02 Pf (G k) = k (k minus 1) (k minus 2)2

= 05 Pf (G k) = k(k minus 1)3

Figure 2 Different fuzzy chromatic polynomials of the fuzzy graph 119866 in Example 23

1

08

07

u

vw

(a) A fuzzy graph 119866 (b) A crisp graph119866120572 for 120572 = 1

Figure 3

Remark 31 For fuzzy graph119866with fuzzy vertex set and fuzzyedge set the degree of the fuzzy chromatic polynomial of 119866is not always equal to the number of vertices of 119866 It can beeasily seen from the fuzzy chromatic polynomial of the fuzzygraph in Example 29

For instance for120572 = 04 the degree of119875119891120572 (119866 119896) = 5whichis the number of vertices of 119866 but for 120572 = 06 the degree of119875119891120572 (119866 119896) = 3 = 5 also for 120572 = 08 the degree of 119875119891120572 (119866 119896) =1 = 5

Remark 32 The fuzzy chromatic polynomial of fuzzy graph119866with fuzzy vertex set and fuzzy edge set 119875119891120572 (119866 119896) 119896 ge 120594120572 willbe decreased when the value of 120572 will increase since in fuzzygraph with fuzzy vertices and fuzzy edges both |V120572| and |E120572|decrease as 120572 increases (see Table 1)

Example 33 Consider a fuzzy graph 119866 with fuzzy vertex setand fuzzy edge set in Figure 6

By routine computation the fuzzy chromatic polynomialof the fuzzy graph 119866 in Figure 6 is obtained as

119875119891120572 (119866 119896) =

k (k minus 1) (k minus 2) 120572 = 0

k (k minus 1) (k minus 2) 120572 = 01

k (k minus 1)2 120572 = 05

k (k minus 1) 120572 = 07

1198962 120572 = 08

119896 120572 = 1

(5)

6 Advances in Fuzzy Systems

Table 1 The values of 120572 |119881120572| |119864120572| 120594120572 and 119875119891120572 (119866 119896) 119896 ge 120594120572 for the fuzzy graph 119866 in Example 33

120572 1003816100381610038161003816V1205721003816100381610038161003816

1003816100381610038161003816E1205721003816100381610038161003816 120594120572 Pf

120572 (G k) Pf120572 (G 120594120572) Pf

120572 (G 120594120572 + 1) Pf120572 (G 120594120572 + 2) Pf

120572 (G 120594120572 + 3)0 3 3 3 k(k-1)(k-2) 6 24 60 12001 3 3 3 k(k-1)(k-2) 6 24 60 12005 3 2 2 k(k-1)2 2 12 36 8007 2 1 2 k(k-1) 2 6 12 2008 2 0 1 k2 1 4 9 161 1 0 1 k 1 2 3 4

02 03

02

03

02

02

03

Ｐ1(04) Ｐ2(06)

Ｐ3(06)

Ｐ4(04)

Ｐ5(08)

Figure 4 A fuzzy graph 119866 with fuzzy vertices and fuzzy edges

Table 1 shows that the fuzzy chromatic polynomial ofthe fuzzy graph 119866 in Example 33 deceases as 120572 increases (seecolumn 1 and columns 6-9 of Table 1) Moreover both thenumber of vertices and the number of edges of 120572-cut graphof the fuzzy graph decrease as 120572 increases (see column 1 andcolumns 2 and 3 of Table 1)

4 Main Results

In this section we prove some relevant results on fuzzychromatic polynomial for fuzzy graphs which are discussedin Section 3 Moreover the fuzzy chromatic polynomial ofcomplete fuzzy graph and the fuzzy chromatic polynomial offuzzy graphs which are cycles and fuzzy cycles are studied

41 120572-Cut Graph and Fuzzy Graph The relations amongdifferent 120572-cuts of a fuzzy graph have been established asfollows

Lemma 34 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then thefollowing are true

(i) |119881120572| ge |119881120573|(ii) |119864120572| ge |119864120573|

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Since0 le 120572 le 120573 le 1 we have 119881120573 sube 119881120572 and 119864120573 sube 119864120572 From thisresults (i) and (ii) hold

Theorem 35 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then119866120573 is a subgraph of 119866120572

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 FromLemma 34 we get 119881120573 sube 119881120572 and 119864120573 sube 119864120572 That means forany element 119906 isin 119881120573 119906 isin 119881120572 and for any element (119906 V) isin 119864120573(119906 V) isin 119864120572Hence 119866120573 is a subgraph of 119866120572

Theorem 36 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 thendeg 119875(119866120572 119896) ge deg 119875(119866120573 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Weknow that the degree of the chromatic polynomial of a crispgraph is the number of vertices in the crisp graph Thereforethe degree of 119875(119866120572 119896) = |119881120572| and the degree of 119875(119866120573 119896) =|119881120573| By Lemma 34(i) we have |119881120572| ge |119881120573| Therefore deg119875(119866120572 119896) = |119881120572| ge |119881120573| = deg 119875(119866120573 119896)

The relations between the 120572-cut graph of a fuzzy graphand the value of 120572 = 0 the fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingcomplete crisp graph can be determined below

Lemma 37 Let 119866 be a fuzzy graph with n vertices and 119866120572 be120572-cut of G en if 120572 = 0 then 119866120572 is a complete crisp graphwith n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0Now1198660 = (1198810 1198640) where1198810 = 119906 isin 119881 | 120590(119906) ge 0 and1198640 = (119906 V) | 120583(119906 V) ge 0 Here 1198810 consists of all the verticesin V of G Similarly 1198640 consists of all the edges in E and allthe edges not in E of G This shows that all the vertices in 1198810of 1198660 are adjacent to each other Therefore 1198660 is a completecrisp graph of n vertices This completes the proof

Remark 38 For 120572-cut of the fuzzy graph 119866 with n vertices119866120572 = 119870119899 if 120572 = 0

Theorem 39 Let119866 be a fuzzy graph with n vertices and119866120572 be120572-cut of G en if 120572 = 0

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (6)

Advances in Fuzzy Systems 7

(k-4)

(k-2)

(k-3)

(k )

(k-1) (k-2)

(k-2)

(k-2)

(k )

(k-1)

( k ) ( k )

( k )

( k )

(k-1) (k-1)

(k-1)

(k )

(k )

(k )

(k ) (k )

(k )

(k )

Ｐ1Ｐ1Ｐ2

Ｐ1 Ｐ2 Ｐ1 Ｐ2

Ｐ2

Ｐ2

Ｐ3 Ｐ3

Ｐ4

Ｐ4 Ｐ4

Ｐ4

Ｐ5

Ｐ3

Ｐ3

Ｐ3

Ｐ5

Ｐ5Ｐ5

Ｐ5

Ｐ5

= 06 Pf (G k) = k3

= 04 Pf (G k) = k5

= 08 Pf (G k) = k

= 03 Pf (G k) = k2 (k minus 1)3

= 02 Pf (G k) = k (k minus 1) (k minus 2)3 = 0 P

f (G k) = k(k minus 1)(k minus 2)(k minus 3)(k minus 4)

Figure 5 Different fuzzy chromatic polynomials of the fuzzy graph G in Example 29

07

0501

w(08)

u(05)

v(1)

Figure 6 A fuzzy graph 119866

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0 Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 119866 is fuzzy graph Thenby Definition 20 and Remark 38 the result holds

The relation between fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingunderlying crisp graph is established below

Theorem 40 Let 119866 be a fuzzy graph and 119866lowast be its underlyingcrisp graph If 120572 = min(119871) where L is the level set orfundamental set of 119866 then 119875119891120572 (119866 119896) = 119875(119866lowast 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and let 119866lowast =(119881lowast 119864lowast) be the underlying crisp graph of119866 where119881lowast = 119906 isin

119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt 0This means that 119881lowast = 119881 and 119864lowast consists of all the edges inE Let 119871 = 1205721 1205722 120572119896 be a level set or fundamental set of119866 Put 120572 = min(119871) now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 120572 isthe minimum value in the level set L 119881120572 consists of all thevertices of V and 119864120572 consists of all the edges in E Therefore119866120572 = 119866lowast Since 119866 is fuzzy graph then by Definition 20 wehave 119875119891120572 (119866 119896) = 119875(119866120572 119896) = 119875(119866lowast 119896)

The following is an important theorem for computing thefuzzy chromatic polynomial of a fuzzy graph explicitly

Theorem 41 Let119866 be a fuzzy graph with n vertices and119866lowast beits underlying crisp graph If 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871)then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119866lowast 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(7)

where 119870119899 is a complete crisp graph with n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices andlet 119866lowast = (119881lowast 119864lowast) be the underlying crisp graph of 119866 where

8 Advances in Fuzzy Systems

119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt0 For 120572 isin I now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theoremby considering three cases

Case 1 Let 120572 = 0 Then by Theorem 39 the fuzzy chromaticpolynomial of the fuzzy graph119866 is the chromatic polynomialof a complete crisp graph with n vertices 119870119899 That is

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (8)

Case 2 Let 120572 = 120573 where 120573 = min(119871) Then by Theorem 40the fuzzy chromatic polynomial of the fuzzy graph 119866 is thechromatic polynomial of its underlying crisp graph 119866lowast Thatis

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (9)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I then by Definition 20119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes the proof

The following result shows that the degree of fuzzychromatic polynomial of a fuzzy graph is less than or equalto the number of vertices of the fuzzy graph It is a significantdifference from crisp graph theory

Theorem 42 Let 119866 = (119881 120590 120583) be a fuzzy graph en thedegree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

Proof Let us prove the theorem by considering two cases

Case 1 Let119866 = (119881 120583) be a fuzzy graph with crisp vertices andfuzzy edges By Lemma 28 the degree of119875119891120572 (119866 119896) = |119881| forall120572 isin119868Therefore the result holds

Case 2 Let 119866 = (119881 120590 120583) be a fuzzy graph with fuzzy vertexset and fuzzy edge set Now 119866120572 = (119881120572 119864120572) where 119881120572 =119906 isin 119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Weknow that the degree of the chromatic polynomial of a crispgraph 119866120572 is the number of vertices of 119866120572 That is the degreeof 119875(119866120572 119896) = |119881120572| But for all 120572 isin 119868 |119881120572| le |119881| since 119881120572 sube VThenbyDefinition 20 we have119875119891120572 (119866 119896) = 119875(119866120572 119896) for120572 isin 119868Therefore the degree of 119875119891120572 (119866 119896) = the degree of 119875(119866120572 119896) =|119881120572| le |119881| This implies that 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868 HencefromCase 1 and Case 2 we conclude that for any fuzzy graph119866 the degree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

42 Complete Fuzzy Graph and Fuzzy Cycle In 1989 Bhutani[11] introduced the concept of a complete fuzzy graph asfollows

Definition 43 (see [11]) A complete fuzzy graph is a fuzzygraph 119866 = (119881 120590 120583) such that 120583(119906 V) = 120590(119906) and 120590(V) for all119906 V isin 119881

The following result gives the underlying crisp graph of acomplete fuzzy graph is a complete crisp graph

Lemma 44 Let 119866 be a complete fuzzy graph and 119866lowast be itsunderlying crisp graph en 119866lowast is a complete crisp graph

It is clear from the following example

Example 45 We show that the underlying crisp graph ofa complete fuzzy graph is complete crisp graph Let 119881 =119906 V 119908 119909 Define the fuzzy set 120590 on V as 120590(119906) = 07 120590(V) =08 120590(119908) = 1 and 120590(119909) = 06 Define a fuzzy set 120583 onE such that 120583(119906 V) = 120583(119906 119908) = 07 120583(119906 119909) = 120583(V 119909) =120583(119908 119909) = 06 and 120583(V 119908) = 08 Then 120583(119906 V) = 120590(119906) and 120590(V)for all 119906 V isin 119881 Thus 119866 = (119881 120590 120583) is a complete fuzzygraph Now119866lowast = (119881lowast 119864lowast) where 119881lowast = 119906 isin 119881120590(119906) gt0 = 119906 V 119908 119909 and 119864lowast = (119906 V) 119906 V isin 119881120583(119906 V) gt0 = (119906 V) (119906 119908) (119906 119909) (V 119909) (119908 119909) (V 119908) This showsthat each vertex in 119866lowast joins each of the other vertices in 119866lowastexactly by one edge Therefore 119866lowast is a complete crisp graph1198704

Remark 46 For complete fuzzy graph 119866 with n vertices byLemma 44 119866lowast = 119870119899 where 119870119899 is a complete crisp graph

The following is an important theorem for computingthe fuzzy chromatic polynomial of a complete fuzzy graphexplicitly

Theorem 47 Let 119866 be a complete fuzzy graph with n verticesIf 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871) then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 and 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1(10)

Proof Let 119866 = (119881 120590 120583) be a complete fuzzy graph with nvertices and let 119866lowast = (119881lowast 119864lowast) be the underlying crisp graphof 119866 where 119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin119881 | 120583(119906 V) gt 0 For 120572 isin I = L cup 0 where L is a level setof 119866 now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen by Theorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (11)

and since 119866 is complete fuzzy graph then by Remark 46 and(11) we get 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes theproof

Remark 48 For a complete fuzzy graph 119866 = (119881 120590 120583) withcrisp vertices (ie 120590(V) = 1 forallV isin 119881) and |119881| = 119899 119875119891120572 (119866 119896) =119875(119870119899 119896) for all 120572 isin I

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

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Advances in Fuzzy Systems 3

Definition 9 The fuzzy graph1198661015840 = (119881 1205901015840 1205831015840) is called a fuzzysubgraph of G if for each two elements u V isin 119881 we have1205901015840(119906) le 120590(119906) and 1205831015840(119906 V) le 120583(119906 V)

Definition 10 The fuzzy graph 119866 = (119881 120590 120583) is calledconnected if for every two elements u V isin 119881 there exists asequence of elements 1199060 1199061 119906119898 such that 1199060 = 119906 119906119898 = Vand 120583(119906119894 119906119894+1) gt 0 (0 le 119894 le 119898 minus 1)

Definition 11 For any fuzzy graph 119866 = (119881 120590 120583) let 119871 =120590(119906) gt 0 | 119906 isin 119881 cup 120583(119906 V) gt 0 | 119906 = V 119906 V isin 119881with 119896 elements Now assume 119871 = 1205721 1205722 120572119896 such that1205721 lt 1205722 lt sdot sdot sdot lt 120572119896 The sequence 1205721 1205722 120572119896 and the set119871 are called the fundamental sequence and the fundamentalset (or level set) of 119866 respectively

Definition 12 The underlying crisp graph 119866lowast = (119881lowast 119864lowast) of afuzzy graph119866 = (119881 120590 120583) is such that119881lowast = 119906 isin 119881 | 120590(119906) gt 0and 119864lowast = (119906 V) | 120583(119906 V) gt 0

Definition 13 For 120572 isin [0 1] 120572-cut graph of a fuzzy graph119866 =(119881 120590 120583) is a crisp graph119866120572 = (119881120572 119864120572) such that119881120572 = 119906 isin 119881 |120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 It is obvious that afuzzy graph will have a finite number of different 120572-cuts

23 Basic Definitions in Fuzzy Coloring of Fuzzy Graphs Theconcept of chromatic number of fuzzy graph was introducedby Munoz et al [22] The authors considered fuzzy graphswith crisp vertex set that is fuzzy graphs for which 120590(V) = 1for all V isin 119881 and edges with membership degree in [0 1]

Definition 14 (see [22]) If 119866 = (119881 120583) is such a fuzzy graphwhere 119881 = 1 2 3119899 and 120583 is a fuzzy number on the set ofall subsets of 119881 times 119881 assume 119868 = 119860 cup 0 where 119860 = 1205721 lt1205722 lt lt 120572119896 is the fundamental set (level set) of 119866 Foreach 120572 isin 119868 119866120572 denote the crisp graph G120572 = (119881 119864120572) where119864120572 = 119894119895|1 le 119894 lt 119895 le 119899 120583(119894 119895) ge 120572 and 120594120572 = 120594(119866120572) denotethe chromatic number of crisp graph 119866120572

By this definition the chromatic number of fuzzy graphsG is the fuzzy number 120594(119866) = (119894 V(119894))119894 isin 119883 where V(119894) =max120572 isin 119868 | 119894 isin 119860120572 and 119860120572 = 1 2 3 120594120572

Later Eslahchi and Onagh [21] introduced fuzzy vertexcoloring of fuzzy graph They defined fuzzy chromatic num-ber as the least value of 119896 for which the fuzzy graph 119866 has119896-fuzzy coloring as follows

Definition 15 (see [21]) A family Γ = 1205741 1205742 120574119896 of fuzzysets on a set 119881 is called a 119896-fuzzy coloring of 119866 = (119881 120590 120583) if

(i) orΓ = 120590

(ii) 120574119894 and 120574119895 = 0

(iii) for every strong edge (119909 119910) (ie 120583(119909 119910) gt 0 ) of 119866min120574119894(119909) 120574119894(119910) = 0 (1 le 119894 le 119896)

Definition 16 (see [21]) The minimum number 119896 for whichthere exists a 119896-fuzzy coloring is called the fuzzy chromaticnumber of 119866 denoted as 120594119891(119866)

Incorporating the features of the above two definitionsKishore and Sunitha [23] modified the chromatic number120594(119866) of fuzzy graph as follows

Definition 17 (see [23]) For each 120572120598119868 119866120572 denote the crispgraph 119866120572 = (120590120572 120583120572) and 120594120572 = 120594(119866120572) denote the chromaticnumber of crisp graph 119866120572 The chromatic number of fuzzygraph 119866 = (119881 120590 120583) is the number 120594(119866) = max120594(119866120572) | 120572 isin119860

3 Fuzzy Chromatic Polynomial ofa Fuzzy Graph

A fuzzy chromatic polynomial is a polynomial which isassociated with the fuzzy coloring of fuzzy graphs There-fore chromatic polynomial in fuzzy graph is called fuzzychromatic polynomial of fuzzy graph In this section wedefine the concept of fuzzy chromatic polynomial of afuzzy graph based on 120572-cuts of fuzzy graph which are crispgraphs Furthermore we determine the fuzzy chromaticpolynomials for some fuzzy graphs with crisp and fuzzyvertices

Crisp vertex coloring and the chromatic polynomial of 120572-cut graph of the fuzzy graph are defined as follows

Definition 18 Let 119866 = (119881 120590 120583) be a fuzzy graph and 119866120572denote120572-cut graph of the fuzzy graph119866which is a crisp graph119866120572 = (119881120572 119864120572) 120572 isin [0 1]A function119862119896120572 119881120572 997888rarr 1 2 119896119896 isin N is called a 119896-coloring (crisp vertex coloring) of 119866120572 if119862119896120572(V119894) = 119862119896120572(V119895) whenever the vertices V119894 and V119895 are adjacentin 119866120572

Definition 19 The number of distinct k-coloring on thevertices of 119866120572 is called the chromatic polynomial of 119866120572 It isdenoted by 119875(119866120572 119896)

Let 119866120572 | 120572 isin 119868 be the family of 120572-cuts sets of 119866where the 120572-cut of a fuzzy graph is the crisp graph119866120572Henceany crisp k-coloring 119862119896120572 can be defined on the vertex set of119866120572 The k-coloring function of the fuzzy graph 119866 is definedthrough this sequence For each 120572 isin 119868 let the fuzzy chromaticpolynomial of 119866 be defined through a monotone family ofsets

Fuzzy chromatic polynomial of a fuzzy graph is definedas follows

Definition 20 Let 119866 be a fuzzy graph The fuzzy chromaticpolynomial of119866 is defined as the chromatic polynomial of itscrisp graphs 119866120572 for 120572 isin 119868 It is denoted by 119875119891120572 (119866 119896)

That is 119875119891120572 (119866 119896) = 119875(119866120572 119896) forall120572 isin 119868

31 Fuzzy Chromatic Polynomial of Fuzzy Graph with CrispVertices In this subsection we present fuzzy chromaticpolynomial of fuzzy graphs with crisp vertices and fuzzyedges

A fuzzy graph 119866 with crisp vertices and fuzzy edges and120572-cut graph of 119866 are defined as follows

4 Advances in Fuzzy Systems

Definition 21 A fuzzy graph is defined as a pair 119866 = (119881 120583)such that

(1) V is the crisp set of vertices (that is120590(119906) = 1 forall119906 isin 119881)(2) the function 120583 119881 x 119881 997888rarr [0 1] is defined by

120583(119906 V) le 120590(119906) and 120590(V) for all 119906 V isin 119881

Definition 22 Let 119866 = (119881 120583) be a fuzzy graph For 120572 isin 119868 120572-cut graph of the fuzzy graph 119866 is defined as the crisp graph119866120572 = (119881 119864120572) where 119864120572 = (119906 V) 119906 V isin 119881 | 120583(119906 V) ge 120572

Example 23 Consider the fuzzy graph 119866 with crisp verticesand fuzzy edges in Figure 1

In 119866 we consider 119868 = 0 02 03 05 08 1 for each120572 isin 119868 we have a crisp graph119866120572 and its chromatic polynomialwhich is the fuzzy chromatic polynomial of the fuzzy graph119866is obtained (see Figure 2) (The integers in the brackets denotethe number of ways of coloring the vertices)

Remark 24 The fuzzy chromatic polynomial depends on thevalues of 120572 which means the fuzzy chromatic polynomialvaries for the same fuzzy graph 119866 for different values of 120572

For the fuzzy graph119866 in Example 23 the fuzzy chromaticpolynomial varies for different values of 120572 as shown below

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0

119896 (119896 minus 1) (119896 minus 2)2 120572 = 02

119896 (119896 minus 1)2 (119896 minus 2) 120572 = 03

119896 (119896 minus 1)3 120572 = 05

1198962 (119896 minus 1)2 120572 = 08

1198964 120572 = 1

(3)

Remark 25 The fuzzy chromatic polynomial of a fuzzy graphdoes not need to be decreased when the value of 120572 increasesLet us consider 1205721 = 02 1205722 = 03 1205723 = 05 1205724 = 08 and1205725 = 1 for the fuzzy graph 119866 in Example 23 but 119875119891120572119894(119866 119896)119896 ge 120594120572119894 where 120594120572119894 is the chromatic number of the crispgraph 119866120572119894 i=12345 is not decreased as 120572119894 increases Forinstance 1198751198911205721(119866 1205941205721) = 6 1198751198911205722(119866 1205941205722) = 12 1198751198911205723(119866 1205941205723) = 21198751198911205724(119866 1205941205724) = 4 and 1198751198911205725(119866 1205941205725) = 1

Remark 26 Let 119866 be a fuzzy graph with n vertices Then for119866 with crisp vertices and fuzzy edges and for 120572 = 1 119875119891120572 (119866 119896)is not always 119875(119873119899 119896) where119873119899 is the null crisp graph withn vertices which is clear from the following example

Example 27 Consider the fuzzy graph119866 given in Figure 3(a)In Figure 3(a) for 120572 = 1 119866120572 is a crisp graph in Figure 3(b)But 119866120572 is not a null crisp graph with three vertices Thus119875119891120572 (119866 119896) = 119875(1198733 119896)

The following result gives the degree of fuzzy chromaticpolynomial of fuzzy graph with crisp vertices and fuzzy edgesare equal to the number of vertices in crisp vertex set

02

030508

08 Ｐ1

Ｐ2Ｐ3

Ｐ4

Figure 1 The fuzzy graph 119866 with crisp vertices and fuzzy edges

Lemma 28 Let119866 = (119881 120583) be a fuzzy graph with crisp verticesand fuzzy edges en the degree of 119875119891120572 (119866 119896) = |119881| for all 120572 isin119868

Proof Let 119866 = (119881 120583) be a fuzzy graph with crisp verticesand fuzzy edges Now 119866120572 = (119881 119864120572) where 119864120572 = (119906 V) |120583(119906 V) ge 120572 We know that the degree of the chromaticpolynomial of a crisp graph 119866120572 is the number of vertices of119866120572 That is the degree of 119875(119866120572 119896) = |119881| By Definition 20we have 119875119891120572 (119866 119896) = 119875(119866120572 119896) for 120572 isin 119868 From this it followsthat the degree of 119875119891120572 (119866 119896) is equal to the degree of 119875(119866120572 119896)Therefore 119875119891120572 (119866 119896) = |119881|

32 Fuzzy Chromatic Polynomial of Fuzzy Graph with FuzzyVertices In Section 31 we introduced the fuzzy chromaticpolynomial of fuzzy graph 119866 = (119881 120583) where 119881 is crispset vertices and 119864 is fuzzy set of edges In this subsectionwe present fuzzy chromatic polynomial of fuzzy graphs withfuzzy vertex set and fuzzy edge set Here we use Definition 8for fuzzy graph and Definition 13 for 120572-cut graph which arepresented in Section 2

Example 29 Consider a fuzzy graph 119866 with fuzzy vertex setand fuzzy edge set in Figure 4

In 119866 we consider 119868 = 0 02 03 04 06 08 for each120572 isin 119868 we have a crisp graph119866120572 and its chromatic polynomialwhich is the fuzzy chromatic polynomial of the fuzzy graph119866is obtained (see Figure 5) (The integers in the brackets denotethe number of ways of coloring the vertices)

Remark 30 The fuzzy chromatic polynomial varies for thesame fuzzy graph 119866 for different values of 120572 It is clear fromExample 29 Here the fuzzy chromatic polynomial of thefuzzy graph in Example 29 is given below

119875119891120572 (119866 119896)

=

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) (119896 minus 4) 120572 = 0

119896 (119896 minus 1) (119896 minus 2)3 120572 = 02

1198962 (119896 minus 1)3 120572 = 03

1198965 120572 = 04

1198963 120572 = 06

119896 120572 = 08

(4)

Advances in Fuzzy Systems 5

(k )

(k-1)(k-2)

(k-1) (k-1)

(k-1)

(k-1)

( k )

(k )

(k-1)(k-2)

(k-2)(k )

(k-1)(k-2)

(k-3)

(k )

(k )

(k )

(k )(k-1) (k-1)

(k )(k )

Ｐ1

Ｐ2

Ｐ2

Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4 Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4Ｐ1

Ｐ3

Ｐ4

= 1 Pf (G k) = k4

= 0 Pf (G k) = k(k minus 1)(k minus 2)(k minus 3)

= 03 Pf (G k) = k (k minus 1)2 (k minus 2)

= 08 Pf (G k) = k2 (k minus 1)2

= 02 Pf (G k) = k (k minus 1) (k minus 2)2

= 05 Pf (G k) = k(k minus 1)3

Figure 2 Different fuzzy chromatic polynomials of the fuzzy graph 119866 in Example 23

1

08

07

u

vw

(a) A fuzzy graph 119866 (b) A crisp graph119866120572 for 120572 = 1

Figure 3

Remark 31 For fuzzy graph119866with fuzzy vertex set and fuzzyedge set the degree of the fuzzy chromatic polynomial of 119866is not always equal to the number of vertices of 119866 It can beeasily seen from the fuzzy chromatic polynomial of the fuzzygraph in Example 29

For instance for120572 = 04 the degree of119875119891120572 (119866 119896) = 5whichis the number of vertices of 119866 but for 120572 = 06 the degree of119875119891120572 (119866 119896) = 3 = 5 also for 120572 = 08 the degree of 119875119891120572 (119866 119896) =1 = 5

Remark 32 The fuzzy chromatic polynomial of fuzzy graph119866with fuzzy vertex set and fuzzy edge set 119875119891120572 (119866 119896) 119896 ge 120594120572 willbe decreased when the value of 120572 will increase since in fuzzygraph with fuzzy vertices and fuzzy edges both |V120572| and |E120572|decrease as 120572 increases (see Table 1)

Example 33 Consider a fuzzy graph 119866 with fuzzy vertex setand fuzzy edge set in Figure 6

By routine computation the fuzzy chromatic polynomialof the fuzzy graph 119866 in Figure 6 is obtained as

119875119891120572 (119866 119896) =

k (k minus 1) (k minus 2) 120572 = 0

k (k minus 1) (k minus 2) 120572 = 01

k (k minus 1)2 120572 = 05

k (k minus 1) 120572 = 07

1198962 120572 = 08

119896 120572 = 1

(5)

6 Advances in Fuzzy Systems

Table 1 The values of 120572 |119881120572| |119864120572| 120594120572 and 119875119891120572 (119866 119896) 119896 ge 120594120572 for the fuzzy graph 119866 in Example 33

120572 1003816100381610038161003816V1205721003816100381610038161003816

1003816100381610038161003816E1205721003816100381610038161003816 120594120572 Pf

120572 (G k) Pf120572 (G 120594120572) Pf

120572 (G 120594120572 + 1) Pf120572 (G 120594120572 + 2) Pf

120572 (G 120594120572 + 3)0 3 3 3 k(k-1)(k-2) 6 24 60 12001 3 3 3 k(k-1)(k-2) 6 24 60 12005 3 2 2 k(k-1)2 2 12 36 8007 2 1 2 k(k-1) 2 6 12 2008 2 0 1 k2 1 4 9 161 1 0 1 k 1 2 3 4

02 03

02

03

02

02

03

Ｐ1(04) Ｐ2(06)

Ｐ3(06)

Ｐ4(04)

Ｐ5(08)

Figure 4 A fuzzy graph 119866 with fuzzy vertices and fuzzy edges

Table 1 shows that the fuzzy chromatic polynomial ofthe fuzzy graph 119866 in Example 33 deceases as 120572 increases (seecolumn 1 and columns 6-9 of Table 1) Moreover both thenumber of vertices and the number of edges of 120572-cut graphof the fuzzy graph decrease as 120572 increases (see column 1 andcolumns 2 and 3 of Table 1)

4 Main Results

In this section we prove some relevant results on fuzzychromatic polynomial for fuzzy graphs which are discussedin Section 3 Moreover the fuzzy chromatic polynomial ofcomplete fuzzy graph and the fuzzy chromatic polynomial offuzzy graphs which are cycles and fuzzy cycles are studied

41 120572-Cut Graph and Fuzzy Graph The relations amongdifferent 120572-cuts of a fuzzy graph have been established asfollows

Lemma 34 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then thefollowing are true

(i) |119881120572| ge |119881120573|(ii) |119864120572| ge |119864120573|

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Since0 le 120572 le 120573 le 1 we have 119881120573 sube 119881120572 and 119864120573 sube 119864120572 From thisresults (i) and (ii) hold

Theorem 35 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then119866120573 is a subgraph of 119866120572

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 FromLemma 34 we get 119881120573 sube 119881120572 and 119864120573 sube 119864120572 That means forany element 119906 isin 119881120573 119906 isin 119881120572 and for any element (119906 V) isin 119864120573(119906 V) isin 119864120572Hence 119866120573 is a subgraph of 119866120572

Theorem 36 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 thendeg 119875(119866120572 119896) ge deg 119875(119866120573 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Weknow that the degree of the chromatic polynomial of a crispgraph is the number of vertices in the crisp graph Thereforethe degree of 119875(119866120572 119896) = |119881120572| and the degree of 119875(119866120573 119896) =|119881120573| By Lemma 34(i) we have |119881120572| ge |119881120573| Therefore deg119875(119866120572 119896) = |119881120572| ge |119881120573| = deg 119875(119866120573 119896)

The relations between the 120572-cut graph of a fuzzy graphand the value of 120572 = 0 the fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingcomplete crisp graph can be determined below

Lemma 37 Let 119866 be a fuzzy graph with n vertices and 119866120572 be120572-cut of G en if 120572 = 0 then 119866120572 is a complete crisp graphwith n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0Now1198660 = (1198810 1198640) where1198810 = 119906 isin 119881 | 120590(119906) ge 0 and1198640 = (119906 V) | 120583(119906 V) ge 0 Here 1198810 consists of all the verticesin V of G Similarly 1198640 consists of all the edges in E and allthe edges not in E of G This shows that all the vertices in 1198810of 1198660 are adjacent to each other Therefore 1198660 is a completecrisp graph of n vertices This completes the proof

Remark 38 For 120572-cut of the fuzzy graph 119866 with n vertices119866120572 = 119870119899 if 120572 = 0

Theorem 39 Let119866 be a fuzzy graph with n vertices and119866120572 be120572-cut of G en if 120572 = 0

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (6)

Advances in Fuzzy Systems 7

(k-4)

(k-2)

(k-3)

(k )

(k-1) (k-2)

(k-2)

(k-2)

(k )

(k-1)

( k ) ( k )

( k )

( k )

(k-1) (k-1)

(k-1)

(k )

(k )

(k )

(k ) (k )

(k )

(k )

Ｐ1Ｐ1Ｐ2

Ｐ1 Ｐ2 Ｐ1 Ｐ2

Ｐ2

Ｐ2

Ｐ3 Ｐ3

Ｐ4

Ｐ4 Ｐ4

Ｐ4

Ｐ5

Ｐ3

Ｐ3

Ｐ3

Ｐ5

Ｐ5Ｐ5

Ｐ5

Ｐ5

= 06 Pf (G k) = k3

= 04 Pf (G k) = k5

= 08 Pf (G k) = k

= 03 Pf (G k) = k2 (k minus 1)3

= 02 Pf (G k) = k (k minus 1) (k minus 2)3 = 0 P

f (G k) = k(k minus 1)(k minus 2)(k minus 3)(k minus 4)

Figure 5 Different fuzzy chromatic polynomials of the fuzzy graph G in Example 29

07

0501

w(08)

u(05)

v(1)

Figure 6 A fuzzy graph 119866

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0 Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 119866 is fuzzy graph Thenby Definition 20 and Remark 38 the result holds

The relation between fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingunderlying crisp graph is established below

Theorem 40 Let 119866 be a fuzzy graph and 119866lowast be its underlyingcrisp graph If 120572 = min(119871) where L is the level set orfundamental set of 119866 then 119875119891120572 (119866 119896) = 119875(119866lowast 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and let 119866lowast =(119881lowast 119864lowast) be the underlying crisp graph of119866 where119881lowast = 119906 isin

119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt 0This means that 119881lowast = 119881 and 119864lowast consists of all the edges inE Let 119871 = 1205721 1205722 120572119896 be a level set or fundamental set of119866 Put 120572 = min(119871) now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 120572 isthe minimum value in the level set L 119881120572 consists of all thevertices of V and 119864120572 consists of all the edges in E Therefore119866120572 = 119866lowast Since 119866 is fuzzy graph then by Definition 20 wehave 119875119891120572 (119866 119896) = 119875(119866120572 119896) = 119875(119866lowast 119896)

The following is an important theorem for computing thefuzzy chromatic polynomial of a fuzzy graph explicitly

Theorem 41 Let119866 be a fuzzy graph with n vertices and119866lowast beits underlying crisp graph If 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871)then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119866lowast 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(7)

where 119870119899 is a complete crisp graph with n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices andlet 119866lowast = (119881lowast 119864lowast) be the underlying crisp graph of 119866 where

8 Advances in Fuzzy Systems

119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt0 For 120572 isin I now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theoremby considering three cases

Case 1 Let 120572 = 0 Then by Theorem 39 the fuzzy chromaticpolynomial of the fuzzy graph119866 is the chromatic polynomialof a complete crisp graph with n vertices 119870119899 That is

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (8)

Case 2 Let 120572 = 120573 where 120573 = min(119871) Then by Theorem 40the fuzzy chromatic polynomial of the fuzzy graph 119866 is thechromatic polynomial of its underlying crisp graph 119866lowast Thatis

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (9)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I then by Definition 20119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes the proof

The following result shows that the degree of fuzzychromatic polynomial of a fuzzy graph is less than or equalto the number of vertices of the fuzzy graph It is a significantdifference from crisp graph theory

Theorem 42 Let 119866 = (119881 120590 120583) be a fuzzy graph en thedegree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

Proof Let us prove the theorem by considering two cases

Case 1 Let119866 = (119881 120583) be a fuzzy graph with crisp vertices andfuzzy edges By Lemma 28 the degree of119875119891120572 (119866 119896) = |119881| forall120572 isin119868Therefore the result holds

Case 2 Let 119866 = (119881 120590 120583) be a fuzzy graph with fuzzy vertexset and fuzzy edge set Now 119866120572 = (119881120572 119864120572) where 119881120572 =119906 isin 119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Weknow that the degree of the chromatic polynomial of a crispgraph 119866120572 is the number of vertices of 119866120572 That is the degreeof 119875(119866120572 119896) = |119881120572| But for all 120572 isin 119868 |119881120572| le |119881| since 119881120572 sube VThenbyDefinition 20 we have119875119891120572 (119866 119896) = 119875(119866120572 119896) for120572 isin 119868Therefore the degree of 119875119891120572 (119866 119896) = the degree of 119875(119866120572 119896) =|119881120572| le |119881| This implies that 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868 HencefromCase 1 and Case 2 we conclude that for any fuzzy graph119866 the degree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

42 Complete Fuzzy Graph and Fuzzy Cycle In 1989 Bhutani[11] introduced the concept of a complete fuzzy graph asfollows

Definition 43 (see [11]) A complete fuzzy graph is a fuzzygraph 119866 = (119881 120590 120583) such that 120583(119906 V) = 120590(119906) and 120590(V) for all119906 V isin 119881

The following result gives the underlying crisp graph of acomplete fuzzy graph is a complete crisp graph

Lemma 44 Let 119866 be a complete fuzzy graph and 119866lowast be itsunderlying crisp graph en 119866lowast is a complete crisp graph

It is clear from the following example

Example 45 We show that the underlying crisp graph ofa complete fuzzy graph is complete crisp graph Let 119881 =119906 V 119908 119909 Define the fuzzy set 120590 on V as 120590(119906) = 07 120590(V) =08 120590(119908) = 1 and 120590(119909) = 06 Define a fuzzy set 120583 onE such that 120583(119906 V) = 120583(119906 119908) = 07 120583(119906 119909) = 120583(V 119909) =120583(119908 119909) = 06 and 120583(V 119908) = 08 Then 120583(119906 V) = 120590(119906) and 120590(V)for all 119906 V isin 119881 Thus 119866 = (119881 120590 120583) is a complete fuzzygraph Now119866lowast = (119881lowast 119864lowast) where 119881lowast = 119906 isin 119881120590(119906) gt0 = 119906 V 119908 119909 and 119864lowast = (119906 V) 119906 V isin 119881120583(119906 V) gt0 = (119906 V) (119906 119908) (119906 119909) (V 119909) (119908 119909) (V 119908) This showsthat each vertex in 119866lowast joins each of the other vertices in 119866lowastexactly by one edge Therefore 119866lowast is a complete crisp graph1198704

Remark 46 For complete fuzzy graph 119866 with n vertices byLemma 44 119866lowast = 119870119899 where 119870119899 is a complete crisp graph

The following is an important theorem for computingthe fuzzy chromatic polynomial of a complete fuzzy graphexplicitly

Theorem 47 Let 119866 be a complete fuzzy graph with n verticesIf 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871) then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 and 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1(10)

Proof Let 119866 = (119881 120590 120583) be a complete fuzzy graph with nvertices and let 119866lowast = (119881lowast 119864lowast) be the underlying crisp graphof 119866 where 119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin119881 | 120583(119906 V) gt 0 For 120572 isin I = L cup 0 where L is a level setof 119866 now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen by Theorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (11)

and since 119866 is complete fuzzy graph then by Remark 46 and(11) we get 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes theproof

Remark 48 For a complete fuzzy graph 119866 = (119881 120590 120583) withcrisp vertices (ie 120590(V) = 1 forallV isin 119881) and |119881| = 119899 119875119891120572 (119866 119896) =119875(119870119899 119896) for all 120572 isin I

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

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Submit your manuscripts atwwwhindawicom

4 Advances in Fuzzy Systems

Definition 21 A fuzzy graph is defined as a pair 119866 = (119881 120583)such that

(1) V is the crisp set of vertices (that is120590(119906) = 1 forall119906 isin 119881)(2) the function 120583 119881 x 119881 997888rarr [0 1] is defined by

120583(119906 V) le 120590(119906) and 120590(V) for all 119906 V isin 119881

Definition 22 Let 119866 = (119881 120583) be a fuzzy graph For 120572 isin 119868 120572-cut graph of the fuzzy graph 119866 is defined as the crisp graph119866120572 = (119881 119864120572) where 119864120572 = (119906 V) 119906 V isin 119881 | 120583(119906 V) ge 120572

Example 23 Consider the fuzzy graph 119866 with crisp verticesand fuzzy edges in Figure 1

In 119866 we consider 119868 = 0 02 03 05 08 1 for each120572 isin 119868 we have a crisp graph119866120572 and its chromatic polynomialwhich is the fuzzy chromatic polynomial of the fuzzy graph119866is obtained (see Figure 2) (The integers in the brackets denotethe number of ways of coloring the vertices)

Remark 24 The fuzzy chromatic polynomial depends on thevalues of 120572 which means the fuzzy chromatic polynomialvaries for the same fuzzy graph 119866 for different values of 120572

For the fuzzy graph119866 in Example 23 the fuzzy chromaticpolynomial varies for different values of 120572 as shown below

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0

119896 (119896 minus 1) (119896 minus 2)2 120572 = 02

119896 (119896 minus 1)2 (119896 minus 2) 120572 = 03

119896 (119896 minus 1)3 120572 = 05

1198962 (119896 minus 1)2 120572 = 08

1198964 120572 = 1

(3)

Remark 25 The fuzzy chromatic polynomial of a fuzzy graphdoes not need to be decreased when the value of 120572 increasesLet us consider 1205721 = 02 1205722 = 03 1205723 = 05 1205724 = 08 and1205725 = 1 for the fuzzy graph 119866 in Example 23 but 119875119891120572119894(119866 119896)119896 ge 120594120572119894 where 120594120572119894 is the chromatic number of the crispgraph 119866120572119894 i=12345 is not decreased as 120572119894 increases Forinstance 1198751198911205721(119866 1205941205721) = 6 1198751198911205722(119866 1205941205722) = 12 1198751198911205723(119866 1205941205723) = 21198751198911205724(119866 1205941205724) = 4 and 1198751198911205725(119866 1205941205725) = 1

Remark 26 Let 119866 be a fuzzy graph with n vertices Then for119866 with crisp vertices and fuzzy edges and for 120572 = 1 119875119891120572 (119866 119896)is not always 119875(119873119899 119896) where119873119899 is the null crisp graph withn vertices which is clear from the following example

Example 27 Consider the fuzzy graph119866 given in Figure 3(a)In Figure 3(a) for 120572 = 1 119866120572 is a crisp graph in Figure 3(b)But 119866120572 is not a null crisp graph with three vertices Thus119875119891120572 (119866 119896) = 119875(1198733 119896)

The following result gives the degree of fuzzy chromaticpolynomial of fuzzy graph with crisp vertices and fuzzy edgesare equal to the number of vertices in crisp vertex set

02

030508

08 Ｐ1

Ｐ2Ｐ3

Ｐ4

Figure 1 The fuzzy graph 119866 with crisp vertices and fuzzy edges

Lemma 28 Let119866 = (119881 120583) be a fuzzy graph with crisp verticesand fuzzy edges en the degree of 119875119891120572 (119866 119896) = |119881| for all 120572 isin119868

Proof Let 119866 = (119881 120583) be a fuzzy graph with crisp verticesand fuzzy edges Now 119866120572 = (119881 119864120572) where 119864120572 = (119906 V) |120583(119906 V) ge 120572 We know that the degree of the chromaticpolynomial of a crisp graph 119866120572 is the number of vertices of119866120572 That is the degree of 119875(119866120572 119896) = |119881| By Definition 20we have 119875119891120572 (119866 119896) = 119875(119866120572 119896) for 120572 isin 119868 From this it followsthat the degree of 119875119891120572 (119866 119896) is equal to the degree of 119875(119866120572 119896)Therefore 119875119891120572 (119866 119896) = |119881|

32 Fuzzy Chromatic Polynomial of Fuzzy Graph with FuzzyVertices In Section 31 we introduced the fuzzy chromaticpolynomial of fuzzy graph 119866 = (119881 120583) where 119881 is crispset vertices and 119864 is fuzzy set of edges In this subsectionwe present fuzzy chromatic polynomial of fuzzy graphs withfuzzy vertex set and fuzzy edge set Here we use Definition 8for fuzzy graph and Definition 13 for 120572-cut graph which arepresented in Section 2

Example 29 Consider a fuzzy graph 119866 with fuzzy vertex setand fuzzy edge set in Figure 4

In 119866 we consider 119868 = 0 02 03 04 06 08 for each120572 isin 119868 we have a crisp graph119866120572 and its chromatic polynomialwhich is the fuzzy chromatic polynomial of the fuzzy graph119866is obtained (see Figure 5) (The integers in the brackets denotethe number of ways of coloring the vertices)

Remark 30 The fuzzy chromatic polynomial varies for thesame fuzzy graph 119866 for different values of 120572 It is clear fromExample 29 Here the fuzzy chromatic polynomial of thefuzzy graph in Example 29 is given below

119875119891120572 (119866 119896)

=

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) (119896 minus 4) 120572 = 0

119896 (119896 minus 1) (119896 minus 2)3 120572 = 02

1198962 (119896 minus 1)3 120572 = 03

1198965 120572 = 04

1198963 120572 = 06

119896 120572 = 08

(4)

Advances in Fuzzy Systems 5

(k )

(k-1)(k-2)

(k-1) (k-1)

(k-1)

(k-1)

( k )

(k )

(k-1)(k-2)

(k-2)(k )

(k-1)(k-2)

(k-3)

(k )

(k )

(k )

(k )(k-1) (k-1)

(k )(k )

Ｐ1

Ｐ2

Ｐ2

Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4 Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4Ｐ1

Ｐ3

Ｐ4

= 1 Pf (G k) = k4

= 0 Pf (G k) = k(k minus 1)(k minus 2)(k minus 3)

= 03 Pf (G k) = k (k minus 1)2 (k minus 2)

= 08 Pf (G k) = k2 (k minus 1)2

= 02 Pf (G k) = k (k minus 1) (k minus 2)2

= 05 Pf (G k) = k(k minus 1)3

Figure 2 Different fuzzy chromatic polynomials of the fuzzy graph 119866 in Example 23

1

08

07

u

vw

(a) A fuzzy graph 119866 (b) A crisp graph119866120572 for 120572 = 1

Figure 3

Remark 31 For fuzzy graph119866with fuzzy vertex set and fuzzyedge set the degree of the fuzzy chromatic polynomial of 119866is not always equal to the number of vertices of 119866 It can beeasily seen from the fuzzy chromatic polynomial of the fuzzygraph in Example 29

For instance for120572 = 04 the degree of119875119891120572 (119866 119896) = 5whichis the number of vertices of 119866 but for 120572 = 06 the degree of119875119891120572 (119866 119896) = 3 = 5 also for 120572 = 08 the degree of 119875119891120572 (119866 119896) =1 = 5

Remark 32 The fuzzy chromatic polynomial of fuzzy graph119866with fuzzy vertex set and fuzzy edge set 119875119891120572 (119866 119896) 119896 ge 120594120572 willbe decreased when the value of 120572 will increase since in fuzzygraph with fuzzy vertices and fuzzy edges both |V120572| and |E120572|decrease as 120572 increases (see Table 1)

Example 33 Consider a fuzzy graph 119866 with fuzzy vertex setand fuzzy edge set in Figure 6

By routine computation the fuzzy chromatic polynomialof the fuzzy graph 119866 in Figure 6 is obtained as

119875119891120572 (119866 119896) =

k (k minus 1) (k minus 2) 120572 = 0

k (k minus 1) (k minus 2) 120572 = 01

k (k minus 1)2 120572 = 05

k (k minus 1) 120572 = 07

1198962 120572 = 08

119896 120572 = 1

(5)

6 Advances in Fuzzy Systems

Table 1 The values of 120572 |119881120572| |119864120572| 120594120572 and 119875119891120572 (119866 119896) 119896 ge 120594120572 for the fuzzy graph 119866 in Example 33

120572 1003816100381610038161003816V1205721003816100381610038161003816

1003816100381610038161003816E1205721003816100381610038161003816 120594120572 Pf

120572 (G k) Pf120572 (G 120594120572) Pf

120572 (G 120594120572 + 1) Pf120572 (G 120594120572 + 2) Pf

120572 (G 120594120572 + 3)0 3 3 3 k(k-1)(k-2) 6 24 60 12001 3 3 3 k(k-1)(k-2) 6 24 60 12005 3 2 2 k(k-1)2 2 12 36 8007 2 1 2 k(k-1) 2 6 12 2008 2 0 1 k2 1 4 9 161 1 0 1 k 1 2 3 4

02 03

02

03

02

02

03

Ｐ1(04) Ｐ2(06)

Ｐ3(06)

Ｐ4(04)

Ｐ5(08)

Figure 4 A fuzzy graph 119866 with fuzzy vertices and fuzzy edges

Table 1 shows that the fuzzy chromatic polynomial ofthe fuzzy graph 119866 in Example 33 deceases as 120572 increases (seecolumn 1 and columns 6-9 of Table 1) Moreover both thenumber of vertices and the number of edges of 120572-cut graphof the fuzzy graph decrease as 120572 increases (see column 1 andcolumns 2 and 3 of Table 1)

4 Main Results

In this section we prove some relevant results on fuzzychromatic polynomial for fuzzy graphs which are discussedin Section 3 Moreover the fuzzy chromatic polynomial ofcomplete fuzzy graph and the fuzzy chromatic polynomial offuzzy graphs which are cycles and fuzzy cycles are studied

41 120572-Cut Graph and Fuzzy Graph The relations amongdifferent 120572-cuts of a fuzzy graph have been established asfollows

Lemma 34 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then thefollowing are true

(i) |119881120572| ge |119881120573|(ii) |119864120572| ge |119864120573|

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Since0 le 120572 le 120573 le 1 we have 119881120573 sube 119881120572 and 119864120573 sube 119864120572 From thisresults (i) and (ii) hold

Theorem 35 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then119866120573 is a subgraph of 119866120572

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 FromLemma 34 we get 119881120573 sube 119881120572 and 119864120573 sube 119864120572 That means forany element 119906 isin 119881120573 119906 isin 119881120572 and for any element (119906 V) isin 119864120573(119906 V) isin 119864120572Hence 119866120573 is a subgraph of 119866120572

Theorem 36 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 thendeg 119875(119866120572 119896) ge deg 119875(119866120573 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Weknow that the degree of the chromatic polynomial of a crispgraph is the number of vertices in the crisp graph Thereforethe degree of 119875(119866120572 119896) = |119881120572| and the degree of 119875(119866120573 119896) =|119881120573| By Lemma 34(i) we have |119881120572| ge |119881120573| Therefore deg119875(119866120572 119896) = |119881120572| ge |119881120573| = deg 119875(119866120573 119896)

The relations between the 120572-cut graph of a fuzzy graphand the value of 120572 = 0 the fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingcomplete crisp graph can be determined below

Lemma 37 Let 119866 be a fuzzy graph with n vertices and 119866120572 be120572-cut of G en if 120572 = 0 then 119866120572 is a complete crisp graphwith n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0Now1198660 = (1198810 1198640) where1198810 = 119906 isin 119881 | 120590(119906) ge 0 and1198640 = (119906 V) | 120583(119906 V) ge 0 Here 1198810 consists of all the verticesin V of G Similarly 1198640 consists of all the edges in E and allthe edges not in E of G This shows that all the vertices in 1198810of 1198660 are adjacent to each other Therefore 1198660 is a completecrisp graph of n vertices This completes the proof

Remark 38 For 120572-cut of the fuzzy graph 119866 with n vertices119866120572 = 119870119899 if 120572 = 0

Theorem 39 Let119866 be a fuzzy graph with n vertices and119866120572 be120572-cut of G en if 120572 = 0

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (6)

Advances in Fuzzy Systems 7

(k-4)

(k-2)

(k-3)

(k )

(k-1) (k-2)

(k-2)

(k-2)

(k )

(k-1)

( k ) ( k )

( k )

( k )

(k-1) (k-1)

(k-1)

(k )

(k )

(k )

(k ) (k )

(k )

(k )

Ｐ1Ｐ1Ｐ2

Ｐ1 Ｐ2 Ｐ1 Ｐ2

Ｐ2

Ｐ2

Ｐ3 Ｐ3

Ｐ4

Ｐ4 Ｐ4

Ｐ4

Ｐ5

Ｐ3

Ｐ3

Ｐ3

Ｐ5

Ｐ5Ｐ5

Ｐ5

Ｐ5

= 06 Pf (G k) = k3

= 04 Pf (G k) = k5

= 08 Pf (G k) = k

= 03 Pf (G k) = k2 (k minus 1)3

= 02 Pf (G k) = k (k minus 1) (k minus 2)3 = 0 P

f (G k) = k(k minus 1)(k minus 2)(k minus 3)(k minus 4)

Figure 5 Different fuzzy chromatic polynomials of the fuzzy graph G in Example 29

07

0501

w(08)

u(05)

v(1)

Figure 6 A fuzzy graph 119866

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0 Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 119866 is fuzzy graph Thenby Definition 20 and Remark 38 the result holds

The relation between fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingunderlying crisp graph is established below

Theorem 40 Let 119866 be a fuzzy graph and 119866lowast be its underlyingcrisp graph If 120572 = min(119871) where L is the level set orfundamental set of 119866 then 119875119891120572 (119866 119896) = 119875(119866lowast 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and let 119866lowast =(119881lowast 119864lowast) be the underlying crisp graph of119866 where119881lowast = 119906 isin

119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt 0This means that 119881lowast = 119881 and 119864lowast consists of all the edges inE Let 119871 = 1205721 1205722 120572119896 be a level set or fundamental set of119866 Put 120572 = min(119871) now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 120572 isthe minimum value in the level set L 119881120572 consists of all thevertices of V and 119864120572 consists of all the edges in E Therefore119866120572 = 119866lowast Since 119866 is fuzzy graph then by Definition 20 wehave 119875119891120572 (119866 119896) = 119875(119866120572 119896) = 119875(119866lowast 119896)

The following is an important theorem for computing thefuzzy chromatic polynomial of a fuzzy graph explicitly

Theorem 41 Let119866 be a fuzzy graph with n vertices and119866lowast beits underlying crisp graph If 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871)then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119866lowast 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(7)

where 119870119899 is a complete crisp graph with n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices andlet 119866lowast = (119881lowast 119864lowast) be the underlying crisp graph of 119866 where

8 Advances in Fuzzy Systems

119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt0 For 120572 isin I now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theoremby considering three cases

Case 1 Let 120572 = 0 Then by Theorem 39 the fuzzy chromaticpolynomial of the fuzzy graph119866 is the chromatic polynomialof a complete crisp graph with n vertices 119870119899 That is

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (8)

Case 2 Let 120572 = 120573 where 120573 = min(119871) Then by Theorem 40the fuzzy chromatic polynomial of the fuzzy graph 119866 is thechromatic polynomial of its underlying crisp graph 119866lowast Thatis

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (9)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I then by Definition 20119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes the proof

The following result shows that the degree of fuzzychromatic polynomial of a fuzzy graph is less than or equalto the number of vertices of the fuzzy graph It is a significantdifference from crisp graph theory

Theorem 42 Let 119866 = (119881 120590 120583) be a fuzzy graph en thedegree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

Proof Let us prove the theorem by considering two cases

Case 1 Let119866 = (119881 120583) be a fuzzy graph with crisp vertices andfuzzy edges By Lemma 28 the degree of119875119891120572 (119866 119896) = |119881| forall120572 isin119868Therefore the result holds

Case 2 Let 119866 = (119881 120590 120583) be a fuzzy graph with fuzzy vertexset and fuzzy edge set Now 119866120572 = (119881120572 119864120572) where 119881120572 =119906 isin 119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Weknow that the degree of the chromatic polynomial of a crispgraph 119866120572 is the number of vertices of 119866120572 That is the degreeof 119875(119866120572 119896) = |119881120572| But for all 120572 isin 119868 |119881120572| le |119881| since 119881120572 sube VThenbyDefinition 20 we have119875119891120572 (119866 119896) = 119875(119866120572 119896) for120572 isin 119868Therefore the degree of 119875119891120572 (119866 119896) = the degree of 119875(119866120572 119896) =|119881120572| le |119881| This implies that 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868 HencefromCase 1 and Case 2 we conclude that for any fuzzy graph119866 the degree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

42 Complete Fuzzy Graph and Fuzzy Cycle In 1989 Bhutani[11] introduced the concept of a complete fuzzy graph asfollows

Definition 43 (see [11]) A complete fuzzy graph is a fuzzygraph 119866 = (119881 120590 120583) such that 120583(119906 V) = 120590(119906) and 120590(V) for all119906 V isin 119881

The following result gives the underlying crisp graph of acomplete fuzzy graph is a complete crisp graph

Lemma 44 Let 119866 be a complete fuzzy graph and 119866lowast be itsunderlying crisp graph en 119866lowast is a complete crisp graph

It is clear from the following example

Example 45 We show that the underlying crisp graph ofa complete fuzzy graph is complete crisp graph Let 119881 =119906 V 119908 119909 Define the fuzzy set 120590 on V as 120590(119906) = 07 120590(V) =08 120590(119908) = 1 and 120590(119909) = 06 Define a fuzzy set 120583 onE such that 120583(119906 V) = 120583(119906 119908) = 07 120583(119906 119909) = 120583(V 119909) =120583(119908 119909) = 06 and 120583(V 119908) = 08 Then 120583(119906 V) = 120590(119906) and 120590(V)for all 119906 V isin 119881 Thus 119866 = (119881 120590 120583) is a complete fuzzygraph Now119866lowast = (119881lowast 119864lowast) where 119881lowast = 119906 isin 119881120590(119906) gt0 = 119906 V 119908 119909 and 119864lowast = (119906 V) 119906 V isin 119881120583(119906 V) gt0 = (119906 V) (119906 119908) (119906 119909) (V 119909) (119908 119909) (V 119908) This showsthat each vertex in 119866lowast joins each of the other vertices in 119866lowastexactly by one edge Therefore 119866lowast is a complete crisp graph1198704

Remark 46 For complete fuzzy graph 119866 with n vertices byLemma 44 119866lowast = 119870119899 where 119870119899 is a complete crisp graph

The following is an important theorem for computingthe fuzzy chromatic polynomial of a complete fuzzy graphexplicitly

Theorem 47 Let 119866 be a complete fuzzy graph with n verticesIf 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871) then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 and 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1(10)

Proof Let 119866 = (119881 120590 120583) be a complete fuzzy graph with nvertices and let 119866lowast = (119881lowast 119864lowast) be the underlying crisp graphof 119866 where 119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin119881 | 120583(119906 V) gt 0 For 120572 isin I = L cup 0 where L is a level setof 119866 now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen by Theorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (11)

and since 119866 is complete fuzzy graph then by Remark 46 and(11) we get 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes theproof

Remark 48 For a complete fuzzy graph 119866 = (119881 120590 120583) withcrisp vertices (ie 120590(V) = 1 forallV isin 119881) and |119881| = 119899 119875119891120572 (119866 119896) =119875(119870119899 119896) for all 120572 isin I

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

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Advances in Fuzzy Systems 5

(k )

(k-1)(k-2)

(k-1) (k-1)

(k-1)

(k-1)

( k )

(k )

(k-1)(k-2)

(k-2)(k )

(k-1)(k-2)

(k-3)

(k )

(k )

(k )

(k )(k-1) (k-1)

(k )(k )

Ｐ1

Ｐ2

Ｐ2

Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4 Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4

Ｐ1

Ｐ2Ｐ3

Ｐ4Ｐ1

Ｐ3

Ｐ4

= 1 Pf (G k) = k4

= 0 Pf (G k) = k(k minus 1)(k minus 2)(k minus 3)

= 03 Pf (G k) = k (k minus 1)2 (k minus 2)

= 08 Pf (G k) = k2 (k minus 1)2

= 02 Pf (G k) = k (k minus 1) (k minus 2)2

= 05 Pf (G k) = k(k minus 1)3

Figure 2 Different fuzzy chromatic polynomials of the fuzzy graph 119866 in Example 23

1

08

07

u

vw

(a) A fuzzy graph 119866 (b) A crisp graph119866120572 for 120572 = 1

Figure 3

Remark 31 For fuzzy graph119866with fuzzy vertex set and fuzzyedge set the degree of the fuzzy chromatic polynomial of 119866is not always equal to the number of vertices of 119866 It can beeasily seen from the fuzzy chromatic polynomial of the fuzzygraph in Example 29

For instance for120572 = 04 the degree of119875119891120572 (119866 119896) = 5whichis the number of vertices of 119866 but for 120572 = 06 the degree of119875119891120572 (119866 119896) = 3 = 5 also for 120572 = 08 the degree of 119875119891120572 (119866 119896) =1 = 5

Remark 32 The fuzzy chromatic polynomial of fuzzy graph119866with fuzzy vertex set and fuzzy edge set 119875119891120572 (119866 119896) 119896 ge 120594120572 willbe decreased when the value of 120572 will increase since in fuzzygraph with fuzzy vertices and fuzzy edges both |V120572| and |E120572|decrease as 120572 increases (see Table 1)

Example 33 Consider a fuzzy graph 119866 with fuzzy vertex setand fuzzy edge set in Figure 6

By routine computation the fuzzy chromatic polynomialof the fuzzy graph 119866 in Figure 6 is obtained as

119875119891120572 (119866 119896) =

k (k minus 1) (k minus 2) 120572 = 0

k (k minus 1) (k minus 2) 120572 = 01

k (k minus 1)2 120572 = 05

k (k minus 1) 120572 = 07

1198962 120572 = 08

119896 120572 = 1

(5)

6 Advances in Fuzzy Systems

Table 1 The values of 120572 |119881120572| |119864120572| 120594120572 and 119875119891120572 (119866 119896) 119896 ge 120594120572 for the fuzzy graph 119866 in Example 33

120572 1003816100381610038161003816V1205721003816100381610038161003816

1003816100381610038161003816E1205721003816100381610038161003816 120594120572 Pf

120572 (G k) Pf120572 (G 120594120572) Pf

120572 (G 120594120572 + 1) Pf120572 (G 120594120572 + 2) Pf

120572 (G 120594120572 + 3)0 3 3 3 k(k-1)(k-2) 6 24 60 12001 3 3 3 k(k-1)(k-2) 6 24 60 12005 3 2 2 k(k-1)2 2 12 36 8007 2 1 2 k(k-1) 2 6 12 2008 2 0 1 k2 1 4 9 161 1 0 1 k 1 2 3 4

02 03

02

03

02

02

03

Ｐ1(04) Ｐ2(06)

Ｐ3(06)

Ｐ4(04)

Ｐ5(08)

Figure 4 A fuzzy graph 119866 with fuzzy vertices and fuzzy edges

Table 1 shows that the fuzzy chromatic polynomial ofthe fuzzy graph 119866 in Example 33 deceases as 120572 increases (seecolumn 1 and columns 6-9 of Table 1) Moreover both thenumber of vertices and the number of edges of 120572-cut graphof the fuzzy graph decrease as 120572 increases (see column 1 andcolumns 2 and 3 of Table 1)

4 Main Results

In this section we prove some relevant results on fuzzychromatic polynomial for fuzzy graphs which are discussedin Section 3 Moreover the fuzzy chromatic polynomial ofcomplete fuzzy graph and the fuzzy chromatic polynomial offuzzy graphs which are cycles and fuzzy cycles are studied

41 120572-Cut Graph and Fuzzy Graph The relations amongdifferent 120572-cuts of a fuzzy graph have been established asfollows

Lemma 34 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then thefollowing are true

(i) |119881120572| ge |119881120573|(ii) |119864120572| ge |119864120573|

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Since0 le 120572 le 120573 le 1 we have 119881120573 sube 119881120572 and 119864120573 sube 119864120572 From thisresults (i) and (ii) hold

Theorem 35 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then119866120573 is a subgraph of 119866120572

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 FromLemma 34 we get 119881120573 sube 119881120572 and 119864120573 sube 119864120572 That means forany element 119906 isin 119881120573 119906 isin 119881120572 and for any element (119906 V) isin 119864120573(119906 V) isin 119864120572Hence 119866120573 is a subgraph of 119866120572

Theorem 36 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 thendeg 119875(119866120572 119896) ge deg 119875(119866120573 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Weknow that the degree of the chromatic polynomial of a crispgraph is the number of vertices in the crisp graph Thereforethe degree of 119875(119866120572 119896) = |119881120572| and the degree of 119875(119866120573 119896) =|119881120573| By Lemma 34(i) we have |119881120572| ge |119881120573| Therefore deg119875(119866120572 119896) = |119881120572| ge |119881120573| = deg 119875(119866120573 119896)

The relations between the 120572-cut graph of a fuzzy graphand the value of 120572 = 0 the fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingcomplete crisp graph can be determined below

Lemma 37 Let 119866 be a fuzzy graph with n vertices and 119866120572 be120572-cut of G en if 120572 = 0 then 119866120572 is a complete crisp graphwith n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0Now1198660 = (1198810 1198640) where1198810 = 119906 isin 119881 | 120590(119906) ge 0 and1198640 = (119906 V) | 120583(119906 V) ge 0 Here 1198810 consists of all the verticesin V of G Similarly 1198640 consists of all the edges in E and allthe edges not in E of G This shows that all the vertices in 1198810of 1198660 are adjacent to each other Therefore 1198660 is a completecrisp graph of n vertices This completes the proof

Remark 38 For 120572-cut of the fuzzy graph 119866 with n vertices119866120572 = 119870119899 if 120572 = 0

Theorem 39 Let119866 be a fuzzy graph with n vertices and119866120572 be120572-cut of G en if 120572 = 0

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (6)

Advances in Fuzzy Systems 7

(k-4)

(k-2)

(k-3)

(k )

(k-1) (k-2)

(k-2)

(k-2)

(k )

(k-1)

( k ) ( k )

( k )

( k )

(k-1) (k-1)

(k-1)

(k )

(k )

(k )

(k ) (k )

(k )

(k )

Ｐ1Ｐ1Ｐ2

Ｐ1 Ｐ2 Ｐ1 Ｐ2

Ｐ2

Ｐ2

Ｐ3 Ｐ3

Ｐ4

Ｐ4 Ｐ4

Ｐ4

Ｐ5

Ｐ3

Ｐ3

Ｐ3

Ｐ5

Ｐ5Ｐ5

Ｐ5

Ｐ5

= 06 Pf (G k) = k3

= 04 Pf (G k) = k5

= 08 Pf (G k) = k

= 03 Pf (G k) = k2 (k minus 1)3

= 02 Pf (G k) = k (k minus 1) (k minus 2)3 = 0 P

f (G k) = k(k minus 1)(k minus 2)(k minus 3)(k minus 4)

Figure 5 Different fuzzy chromatic polynomials of the fuzzy graph G in Example 29

07

0501

w(08)

u(05)

v(1)

Figure 6 A fuzzy graph 119866

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0 Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 119866 is fuzzy graph Thenby Definition 20 and Remark 38 the result holds

The relation between fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingunderlying crisp graph is established below

Theorem 40 Let 119866 be a fuzzy graph and 119866lowast be its underlyingcrisp graph If 120572 = min(119871) where L is the level set orfundamental set of 119866 then 119875119891120572 (119866 119896) = 119875(119866lowast 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and let 119866lowast =(119881lowast 119864lowast) be the underlying crisp graph of119866 where119881lowast = 119906 isin

119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt 0This means that 119881lowast = 119881 and 119864lowast consists of all the edges inE Let 119871 = 1205721 1205722 120572119896 be a level set or fundamental set of119866 Put 120572 = min(119871) now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 120572 isthe minimum value in the level set L 119881120572 consists of all thevertices of V and 119864120572 consists of all the edges in E Therefore119866120572 = 119866lowast Since 119866 is fuzzy graph then by Definition 20 wehave 119875119891120572 (119866 119896) = 119875(119866120572 119896) = 119875(119866lowast 119896)

The following is an important theorem for computing thefuzzy chromatic polynomial of a fuzzy graph explicitly

Theorem 41 Let119866 be a fuzzy graph with n vertices and119866lowast beits underlying crisp graph If 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871)then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119866lowast 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(7)

where 119870119899 is a complete crisp graph with n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices andlet 119866lowast = (119881lowast 119864lowast) be the underlying crisp graph of 119866 where

8 Advances in Fuzzy Systems

119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt0 For 120572 isin I now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theoremby considering three cases

Case 1 Let 120572 = 0 Then by Theorem 39 the fuzzy chromaticpolynomial of the fuzzy graph119866 is the chromatic polynomialof a complete crisp graph with n vertices 119870119899 That is

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (8)

Case 2 Let 120572 = 120573 where 120573 = min(119871) Then by Theorem 40the fuzzy chromatic polynomial of the fuzzy graph 119866 is thechromatic polynomial of its underlying crisp graph 119866lowast Thatis

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (9)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I then by Definition 20119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes the proof

The following result shows that the degree of fuzzychromatic polynomial of a fuzzy graph is less than or equalto the number of vertices of the fuzzy graph It is a significantdifference from crisp graph theory

Theorem 42 Let 119866 = (119881 120590 120583) be a fuzzy graph en thedegree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

Proof Let us prove the theorem by considering two cases

Case 1 Let119866 = (119881 120583) be a fuzzy graph with crisp vertices andfuzzy edges By Lemma 28 the degree of119875119891120572 (119866 119896) = |119881| forall120572 isin119868Therefore the result holds

Case 2 Let 119866 = (119881 120590 120583) be a fuzzy graph with fuzzy vertexset and fuzzy edge set Now 119866120572 = (119881120572 119864120572) where 119881120572 =119906 isin 119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Weknow that the degree of the chromatic polynomial of a crispgraph 119866120572 is the number of vertices of 119866120572 That is the degreeof 119875(119866120572 119896) = |119881120572| But for all 120572 isin 119868 |119881120572| le |119881| since 119881120572 sube VThenbyDefinition 20 we have119875119891120572 (119866 119896) = 119875(119866120572 119896) for120572 isin 119868Therefore the degree of 119875119891120572 (119866 119896) = the degree of 119875(119866120572 119896) =|119881120572| le |119881| This implies that 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868 HencefromCase 1 and Case 2 we conclude that for any fuzzy graph119866 the degree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

42 Complete Fuzzy Graph and Fuzzy Cycle In 1989 Bhutani[11] introduced the concept of a complete fuzzy graph asfollows

Definition 43 (see [11]) A complete fuzzy graph is a fuzzygraph 119866 = (119881 120590 120583) such that 120583(119906 V) = 120590(119906) and 120590(V) for all119906 V isin 119881

The following result gives the underlying crisp graph of acomplete fuzzy graph is a complete crisp graph

Lemma 44 Let 119866 be a complete fuzzy graph and 119866lowast be itsunderlying crisp graph en 119866lowast is a complete crisp graph

It is clear from the following example

Example 45 We show that the underlying crisp graph ofa complete fuzzy graph is complete crisp graph Let 119881 =119906 V 119908 119909 Define the fuzzy set 120590 on V as 120590(119906) = 07 120590(V) =08 120590(119908) = 1 and 120590(119909) = 06 Define a fuzzy set 120583 onE such that 120583(119906 V) = 120583(119906 119908) = 07 120583(119906 119909) = 120583(V 119909) =120583(119908 119909) = 06 and 120583(V 119908) = 08 Then 120583(119906 V) = 120590(119906) and 120590(V)for all 119906 V isin 119881 Thus 119866 = (119881 120590 120583) is a complete fuzzygraph Now119866lowast = (119881lowast 119864lowast) where 119881lowast = 119906 isin 119881120590(119906) gt0 = 119906 V 119908 119909 and 119864lowast = (119906 V) 119906 V isin 119881120583(119906 V) gt0 = (119906 V) (119906 119908) (119906 119909) (V 119909) (119908 119909) (V 119908) This showsthat each vertex in 119866lowast joins each of the other vertices in 119866lowastexactly by one edge Therefore 119866lowast is a complete crisp graph1198704

Remark 46 For complete fuzzy graph 119866 with n vertices byLemma 44 119866lowast = 119870119899 where 119870119899 is a complete crisp graph

The following is an important theorem for computingthe fuzzy chromatic polynomial of a complete fuzzy graphexplicitly

Theorem 47 Let 119866 be a complete fuzzy graph with n verticesIf 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871) then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 and 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1(10)

Proof Let 119866 = (119881 120590 120583) be a complete fuzzy graph with nvertices and let 119866lowast = (119881lowast 119864lowast) be the underlying crisp graphof 119866 where 119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin119881 | 120583(119906 V) gt 0 For 120572 isin I = L cup 0 where L is a level setof 119866 now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen by Theorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (11)

and since 119866 is complete fuzzy graph then by Remark 46 and(11) we get 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes theproof

Remark 48 For a complete fuzzy graph 119866 = (119881 120590 120583) withcrisp vertices (ie 120590(V) = 1 forallV isin 119881) and |119881| = 119899 119875119891120572 (119866 119896) =119875(119870119899 119896) for all 120572 isin I

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

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Advances in

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Submit your manuscripts atwwwhindawicom

6 Advances in Fuzzy Systems

Table 1 The values of 120572 |119881120572| |119864120572| 120594120572 and 119875119891120572 (119866 119896) 119896 ge 120594120572 for the fuzzy graph 119866 in Example 33

120572 1003816100381610038161003816V1205721003816100381610038161003816

1003816100381610038161003816E1205721003816100381610038161003816 120594120572 Pf

120572 (G k) Pf120572 (G 120594120572) Pf

120572 (G 120594120572 + 1) Pf120572 (G 120594120572 + 2) Pf

120572 (G 120594120572 + 3)0 3 3 3 k(k-1)(k-2) 6 24 60 12001 3 3 3 k(k-1)(k-2) 6 24 60 12005 3 2 2 k(k-1)2 2 12 36 8007 2 1 2 k(k-1) 2 6 12 2008 2 0 1 k2 1 4 9 161 1 0 1 k 1 2 3 4

02 03

02

03

02

02

03

Ｐ1(04) Ｐ2(06)

Ｐ3(06)

Ｐ4(04)

Ｐ5(08)

Figure 4 A fuzzy graph 119866 with fuzzy vertices and fuzzy edges

Table 1 shows that the fuzzy chromatic polynomial ofthe fuzzy graph 119866 in Example 33 deceases as 120572 increases (seecolumn 1 and columns 6-9 of Table 1) Moreover both thenumber of vertices and the number of edges of 120572-cut graphof the fuzzy graph decrease as 120572 increases (see column 1 andcolumns 2 and 3 of Table 1)

4 Main Results

In this section we prove some relevant results on fuzzychromatic polynomial for fuzzy graphs which are discussedin Section 3 Moreover the fuzzy chromatic polynomial ofcomplete fuzzy graph and the fuzzy chromatic polynomial offuzzy graphs which are cycles and fuzzy cycles are studied

41 120572-Cut Graph and Fuzzy Graph The relations amongdifferent 120572-cuts of a fuzzy graph have been established asfollows

Lemma 34 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then thefollowing are true

(i) |119881120572| ge |119881120573|(ii) |119864120572| ge |119864120573|

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Since0 le 120572 le 120573 le 1 we have 119881120573 sube 119881120572 and 119864120573 sube 119864120572 From thisresults (i) and (ii) hold

Theorem 35 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 then119866120573 is a subgraph of 119866120572

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 FromLemma 34 we get 119881120573 sube 119881120572 and 119864120573 sube 119864120572 That means forany element 119906 isin 119881120573 119906 isin 119881120572 and for any element (119906 V) isin 119864120573(119906 V) isin 119864120572Hence 119866120573 is a subgraph of 119866120572

Theorem 36 Let 119866 be a fuzzy graph If 0 le 120572 le 120573 le 1 thendeg 119875(119866120572 119896) ge deg 119875(119866120573 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and 0 le 120572 le 120573 le 1Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Also 119866120573 = (119881120573 119864120573) where 119881120573 =119906 isin 119881 | 120590(119906) ge 120573 and 119864120573 = (119906 V) | 120583(119906 V) ge 120573 Weknow that the degree of the chromatic polynomial of a crispgraph is the number of vertices in the crisp graph Thereforethe degree of 119875(119866120572 119896) = |119881120572| and the degree of 119875(119866120573 119896) =|119881120573| By Lemma 34(i) we have |119881120572| ge |119881120573| Therefore deg119875(119866120572 119896) = |119881120572| ge |119881120573| = deg 119875(119866120573 119896)

The relations between the 120572-cut graph of a fuzzy graphand the value of 120572 = 0 the fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingcomplete crisp graph can be determined below

Lemma 37 Let 119866 be a fuzzy graph with n vertices and 119866120572 be120572-cut of G en if 120572 = 0 then 119866120572 is a complete crisp graphwith n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0Now1198660 = (1198810 1198640) where1198810 = 119906 isin 119881 | 120590(119906) ge 0 and1198640 = (119906 V) | 120583(119906 V) ge 0 Here 1198810 consists of all the verticesin V of G Similarly 1198640 consists of all the edges in E and allthe edges not in E of G This shows that all the vertices in 1198810of 1198660 are adjacent to each other Therefore 1198660 is a completecrisp graph of n vertices This completes the proof

Remark 38 For 120572-cut of the fuzzy graph 119866 with n vertices119866120572 = 119870119899 if 120572 = 0

Theorem 39 Let119866 be a fuzzy graph with n vertices and119866120572 be120572-cut of G en if 120572 = 0

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (6)

Advances in Fuzzy Systems 7

(k-4)

(k-2)

(k-3)

(k )

(k-1) (k-2)

(k-2)

(k-2)

(k )

(k-1)

( k ) ( k )

( k )

( k )

(k-1) (k-1)

(k-1)

(k )

(k )

(k )

(k ) (k )

(k )

(k )

Ｐ1Ｐ1Ｐ2

Ｐ1 Ｐ2 Ｐ1 Ｐ2

Ｐ2

Ｐ2

Ｐ3 Ｐ3

Ｐ4

Ｐ4 Ｐ4

Ｐ4

Ｐ5

Ｐ3

Ｐ3

Ｐ3

Ｐ5

Ｐ5Ｐ5

Ｐ5

Ｐ5

= 06 Pf (G k) = k3

= 04 Pf (G k) = k5

= 08 Pf (G k) = k

= 03 Pf (G k) = k2 (k minus 1)3

= 02 Pf (G k) = k (k minus 1) (k minus 2)3 = 0 P

f (G k) = k(k minus 1)(k minus 2)(k minus 3)(k minus 4)

Figure 5 Different fuzzy chromatic polynomials of the fuzzy graph G in Example 29

07

0501

w(08)

u(05)

v(1)

Figure 6 A fuzzy graph 119866

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0 Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 119866 is fuzzy graph Thenby Definition 20 and Remark 38 the result holds

The relation between fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingunderlying crisp graph is established below

Theorem 40 Let 119866 be a fuzzy graph and 119866lowast be its underlyingcrisp graph If 120572 = min(119871) where L is the level set orfundamental set of 119866 then 119875119891120572 (119866 119896) = 119875(119866lowast 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and let 119866lowast =(119881lowast 119864lowast) be the underlying crisp graph of119866 where119881lowast = 119906 isin

119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt 0This means that 119881lowast = 119881 and 119864lowast consists of all the edges inE Let 119871 = 1205721 1205722 120572119896 be a level set or fundamental set of119866 Put 120572 = min(119871) now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 120572 isthe minimum value in the level set L 119881120572 consists of all thevertices of V and 119864120572 consists of all the edges in E Therefore119866120572 = 119866lowast Since 119866 is fuzzy graph then by Definition 20 wehave 119875119891120572 (119866 119896) = 119875(119866120572 119896) = 119875(119866lowast 119896)

The following is an important theorem for computing thefuzzy chromatic polynomial of a fuzzy graph explicitly

Theorem 41 Let119866 be a fuzzy graph with n vertices and119866lowast beits underlying crisp graph If 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871)then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119866lowast 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(7)

where 119870119899 is a complete crisp graph with n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices andlet 119866lowast = (119881lowast 119864lowast) be the underlying crisp graph of 119866 where

8 Advances in Fuzzy Systems

119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt0 For 120572 isin I now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theoremby considering three cases

Case 1 Let 120572 = 0 Then by Theorem 39 the fuzzy chromaticpolynomial of the fuzzy graph119866 is the chromatic polynomialof a complete crisp graph with n vertices 119870119899 That is

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (8)

Case 2 Let 120572 = 120573 where 120573 = min(119871) Then by Theorem 40the fuzzy chromatic polynomial of the fuzzy graph 119866 is thechromatic polynomial of its underlying crisp graph 119866lowast Thatis

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (9)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I then by Definition 20119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes the proof

The following result shows that the degree of fuzzychromatic polynomial of a fuzzy graph is less than or equalto the number of vertices of the fuzzy graph It is a significantdifference from crisp graph theory

Theorem 42 Let 119866 = (119881 120590 120583) be a fuzzy graph en thedegree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

Proof Let us prove the theorem by considering two cases

Case 1 Let119866 = (119881 120583) be a fuzzy graph with crisp vertices andfuzzy edges By Lemma 28 the degree of119875119891120572 (119866 119896) = |119881| forall120572 isin119868Therefore the result holds

Case 2 Let 119866 = (119881 120590 120583) be a fuzzy graph with fuzzy vertexset and fuzzy edge set Now 119866120572 = (119881120572 119864120572) where 119881120572 =119906 isin 119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Weknow that the degree of the chromatic polynomial of a crispgraph 119866120572 is the number of vertices of 119866120572 That is the degreeof 119875(119866120572 119896) = |119881120572| But for all 120572 isin 119868 |119881120572| le |119881| since 119881120572 sube VThenbyDefinition 20 we have119875119891120572 (119866 119896) = 119875(119866120572 119896) for120572 isin 119868Therefore the degree of 119875119891120572 (119866 119896) = the degree of 119875(119866120572 119896) =|119881120572| le |119881| This implies that 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868 HencefromCase 1 and Case 2 we conclude that for any fuzzy graph119866 the degree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

42 Complete Fuzzy Graph and Fuzzy Cycle In 1989 Bhutani[11] introduced the concept of a complete fuzzy graph asfollows

Definition 43 (see [11]) A complete fuzzy graph is a fuzzygraph 119866 = (119881 120590 120583) such that 120583(119906 V) = 120590(119906) and 120590(V) for all119906 V isin 119881

The following result gives the underlying crisp graph of acomplete fuzzy graph is a complete crisp graph

Lemma 44 Let 119866 be a complete fuzzy graph and 119866lowast be itsunderlying crisp graph en 119866lowast is a complete crisp graph

It is clear from the following example

Example 45 We show that the underlying crisp graph ofa complete fuzzy graph is complete crisp graph Let 119881 =119906 V 119908 119909 Define the fuzzy set 120590 on V as 120590(119906) = 07 120590(V) =08 120590(119908) = 1 and 120590(119909) = 06 Define a fuzzy set 120583 onE such that 120583(119906 V) = 120583(119906 119908) = 07 120583(119906 119909) = 120583(V 119909) =120583(119908 119909) = 06 and 120583(V 119908) = 08 Then 120583(119906 V) = 120590(119906) and 120590(V)for all 119906 V isin 119881 Thus 119866 = (119881 120590 120583) is a complete fuzzygraph Now119866lowast = (119881lowast 119864lowast) where 119881lowast = 119906 isin 119881120590(119906) gt0 = 119906 V 119908 119909 and 119864lowast = (119906 V) 119906 V isin 119881120583(119906 V) gt0 = (119906 V) (119906 119908) (119906 119909) (V 119909) (119908 119909) (V 119908) This showsthat each vertex in 119866lowast joins each of the other vertices in 119866lowastexactly by one edge Therefore 119866lowast is a complete crisp graph1198704

Remark 46 For complete fuzzy graph 119866 with n vertices byLemma 44 119866lowast = 119870119899 where 119870119899 is a complete crisp graph

The following is an important theorem for computingthe fuzzy chromatic polynomial of a complete fuzzy graphexplicitly

Theorem 47 Let 119866 be a complete fuzzy graph with n verticesIf 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871) then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 and 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1(10)

Proof Let 119866 = (119881 120590 120583) be a complete fuzzy graph with nvertices and let 119866lowast = (119881lowast 119864lowast) be the underlying crisp graphof 119866 where 119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin119881 | 120583(119906 V) gt 0 For 120572 isin I = L cup 0 where L is a level setof 119866 now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen by Theorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (11)

and since 119866 is complete fuzzy graph then by Remark 46 and(11) we get 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes theproof

Remark 48 For a complete fuzzy graph 119866 = (119881 120590 120583) withcrisp vertices (ie 120590(V) = 1 forallV isin 119881) and |119881| = 119899 119875119891120572 (119866 119896) =119875(119870119899 119896) for all 120572 isin I

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

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Submit your manuscripts atwwwhindawicom

Advances in Fuzzy Systems 7

(k-4)

(k-2)

(k-3)

(k )

(k-1) (k-2)

(k-2)

(k-2)

(k )

(k-1)

( k ) ( k )

( k )

( k )

(k-1) (k-1)

(k-1)

(k )

(k )

(k )

(k ) (k )

(k )

(k )

Ｐ1Ｐ1Ｐ2

Ｐ1 Ｐ2 Ｐ1 Ｐ2

Ｐ2

Ｐ2

Ｐ3 Ｐ3

Ｐ4

Ｐ4 Ｐ4

Ｐ4

Ｐ5

Ｐ3

Ｐ3

Ｐ3

Ｐ5

Ｐ5Ｐ5

Ｐ5

Ｐ5

= 06 Pf (G k) = k3

= 04 Pf (G k) = k5

= 08 Pf (G k) = k

= 03 Pf (G k) = k2 (k minus 1)3

= 02 Pf (G k) = k (k minus 1) (k minus 2)3 = 0 P

f (G k) = k(k minus 1)(k minus 2)(k minus 3)(k minus 4)

Figure 5 Different fuzzy chromatic polynomials of the fuzzy graph G in Example 29

07

0501

w(08)

u(05)

v(1)

Figure 6 A fuzzy graph 119866

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices and120572 = 0 Now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 119866 is fuzzy graph Thenby Definition 20 and Remark 38 the result holds

The relation between fuzzy chromatic polynomial of afuzzy graph and the chromatic polynomial of correspondingunderlying crisp graph is established below

Theorem 40 Let 119866 be a fuzzy graph and 119866lowast be its underlyingcrisp graph If 120572 = min(119871) where L is the level set orfundamental set of 119866 then 119875119891120572 (119866 119896) = 119875(119866lowast 119896)

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph and let 119866lowast =(119881lowast 119864lowast) be the underlying crisp graph of119866 where119881lowast = 119906 isin

119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt 0This means that 119881lowast = 119881 and 119864lowast consists of all the edges inE Let 119871 = 1205721 1205722 120572119896 be a level set or fundamental set of119866 Put 120572 = min(119871) now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Since 120572 isthe minimum value in the level set L 119881120572 consists of all thevertices of V and 119864120572 consists of all the edges in E Therefore119866120572 = 119866lowast Since 119866 is fuzzy graph then by Definition 20 wehave 119875119891120572 (119866 119896) = 119875(119866120572 119896) = 119875(119866lowast 119896)

The following is an important theorem for computing thefuzzy chromatic polynomial of a fuzzy graph explicitly

Theorem 41 Let119866 be a fuzzy graph with n vertices and119866lowast beits underlying crisp graph If 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871)then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119866lowast 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(7)

where 119870119899 is a complete crisp graph with n vertices

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n vertices andlet 119866lowast = (119881lowast 119864lowast) be the underlying crisp graph of 119866 where

8 Advances in Fuzzy Systems

119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt0 For 120572 isin I now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theoremby considering three cases

Case 1 Let 120572 = 0 Then by Theorem 39 the fuzzy chromaticpolynomial of the fuzzy graph119866 is the chromatic polynomialof a complete crisp graph with n vertices 119870119899 That is

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (8)

Case 2 Let 120572 = 120573 where 120573 = min(119871) Then by Theorem 40the fuzzy chromatic polynomial of the fuzzy graph 119866 is thechromatic polynomial of its underlying crisp graph 119866lowast Thatis

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (9)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I then by Definition 20119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes the proof

The following result shows that the degree of fuzzychromatic polynomial of a fuzzy graph is less than or equalto the number of vertices of the fuzzy graph It is a significantdifference from crisp graph theory

Theorem 42 Let 119866 = (119881 120590 120583) be a fuzzy graph en thedegree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

Proof Let us prove the theorem by considering two cases

Case 1 Let119866 = (119881 120583) be a fuzzy graph with crisp vertices andfuzzy edges By Lemma 28 the degree of119875119891120572 (119866 119896) = |119881| forall120572 isin119868Therefore the result holds

Case 2 Let 119866 = (119881 120590 120583) be a fuzzy graph with fuzzy vertexset and fuzzy edge set Now 119866120572 = (119881120572 119864120572) where 119881120572 =119906 isin 119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Weknow that the degree of the chromatic polynomial of a crispgraph 119866120572 is the number of vertices of 119866120572 That is the degreeof 119875(119866120572 119896) = |119881120572| But for all 120572 isin 119868 |119881120572| le |119881| since 119881120572 sube VThenbyDefinition 20 we have119875119891120572 (119866 119896) = 119875(119866120572 119896) for120572 isin 119868Therefore the degree of 119875119891120572 (119866 119896) = the degree of 119875(119866120572 119896) =|119881120572| le |119881| This implies that 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868 HencefromCase 1 and Case 2 we conclude that for any fuzzy graph119866 the degree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

42 Complete Fuzzy Graph and Fuzzy Cycle In 1989 Bhutani[11] introduced the concept of a complete fuzzy graph asfollows

Definition 43 (see [11]) A complete fuzzy graph is a fuzzygraph 119866 = (119881 120590 120583) such that 120583(119906 V) = 120590(119906) and 120590(V) for all119906 V isin 119881

The following result gives the underlying crisp graph of acomplete fuzzy graph is a complete crisp graph

Lemma 44 Let 119866 be a complete fuzzy graph and 119866lowast be itsunderlying crisp graph en 119866lowast is a complete crisp graph

It is clear from the following example

Example 45 We show that the underlying crisp graph ofa complete fuzzy graph is complete crisp graph Let 119881 =119906 V 119908 119909 Define the fuzzy set 120590 on V as 120590(119906) = 07 120590(V) =08 120590(119908) = 1 and 120590(119909) = 06 Define a fuzzy set 120583 onE such that 120583(119906 V) = 120583(119906 119908) = 07 120583(119906 119909) = 120583(V 119909) =120583(119908 119909) = 06 and 120583(V 119908) = 08 Then 120583(119906 V) = 120590(119906) and 120590(V)for all 119906 V isin 119881 Thus 119866 = (119881 120590 120583) is a complete fuzzygraph Now119866lowast = (119881lowast 119864lowast) where 119881lowast = 119906 isin 119881120590(119906) gt0 = 119906 V 119908 119909 and 119864lowast = (119906 V) 119906 V isin 119881120583(119906 V) gt0 = (119906 V) (119906 119908) (119906 119909) (V 119909) (119908 119909) (V 119908) This showsthat each vertex in 119866lowast joins each of the other vertices in 119866lowastexactly by one edge Therefore 119866lowast is a complete crisp graph1198704

Remark 46 For complete fuzzy graph 119866 with n vertices byLemma 44 119866lowast = 119870119899 where 119870119899 is a complete crisp graph

The following is an important theorem for computingthe fuzzy chromatic polynomial of a complete fuzzy graphexplicitly

Theorem 47 Let 119866 be a complete fuzzy graph with n verticesIf 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871) then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 and 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1(10)

Proof Let 119866 = (119881 120590 120583) be a complete fuzzy graph with nvertices and let 119866lowast = (119881lowast 119864lowast) be the underlying crisp graphof 119866 where 119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin119881 | 120583(119906 V) gt 0 For 120572 isin I = L cup 0 where L is a level setof 119866 now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen by Theorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (11)

and since 119866 is complete fuzzy graph then by Remark 46 and(11) we get 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes theproof

Remark 48 For a complete fuzzy graph 119866 = (119881 120590 120583) withcrisp vertices (ie 120590(V) = 1 forallV isin 119881) and |119881| = 119899 119875119891120572 (119866 119896) =119875(119870119899 119896) for all 120572 isin I

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

8 Advances in Fuzzy Systems

119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin 119881 | 120583(119906 V) gt0 For 120572 isin I now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theoremby considering three cases

Case 1 Let 120572 = 0 Then by Theorem 39 the fuzzy chromaticpolynomial of the fuzzy graph119866 is the chromatic polynomialof a complete crisp graph with n vertices 119870119899 That is

119875119891120572 (119866 119896) = 119875 (119870119899 119896) (8)

Case 2 Let 120572 = 120573 where 120573 = min(119871) Then by Theorem 40the fuzzy chromatic polynomial of the fuzzy graph 119866 is thechromatic polynomial of its underlying crisp graph 119866lowast Thatis

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (9)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I then by Definition 20119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes the proof

The following result shows that the degree of fuzzychromatic polynomial of a fuzzy graph is less than or equalto the number of vertices of the fuzzy graph It is a significantdifference from crisp graph theory

Theorem 42 Let 119866 = (119881 120590 120583) be a fuzzy graph en thedegree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

Proof Let us prove the theorem by considering two cases

Case 1 Let119866 = (119881 120583) be a fuzzy graph with crisp vertices andfuzzy edges By Lemma 28 the degree of119875119891120572 (119866 119896) = |119881| forall120572 isin119868Therefore the result holds

Case 2 Let 119866 = (119881 120590 120583) be a fuzzy graph with fuzzy vertexset and fuzzy edge set Now 119866120572 = (119881120572 119864120572) where 119881120572 =119906 isin 119881 | 120590(119906) ge 120572 and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Weknow that the degree of the chromatic polynomial of a crispgraph 119866120572 is the number of vertices of 119866120572 That is the degreeof 119875(119866120572 119896) = |119881120572| But for all 120572 isin 119868 |119881120572| le |119881| since 119881120572 sube VThenbyDefinition 20 we have119875119891120572 (119866 119896) = 119875(119866120572 119896) for120572 isin 119868Therefore the degree of 119875119891120572 (119866 119896) = the degree of 119875(119866120572 119896) =|119881120572| le |119881| This implies that 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868 HencefromCase 1 and Case 2 we conclude that for any fuzzy graph119866 the degree of 119875119891120572 (119866 119896) le |119881| forall120572 isin 119868

42 Complete Fuzzy Graph and Fuzzy Cycle In 1989 Bhutani[11] introduced the concept of a complete fuzzy graph asfollows

Definition 43 (see [11]) A complete fuzzy graph is a fuzzygraph 119866 = (119881 120590 120583) such that 120583(119906 V) = 120590(119906) and 120590(V) for all119906 V isin 119881

The following result gives the underlying crisp graph of acomplete fuzzy graph is a complete crisp graph

Lemma 44 Let 119866 be a complete fuzzy graph and 119866lowast be itsunderlying crisp graph en 119866lowast is a complete crisp graph

It is clear from the following example

Example 45 We show that the underlying crisp graph ofa complete fuzzy graph is complete crisp graph Let 119881 =119906 V 119908 119909 Define the fuzzy set 120590 on V as 120590(119906) = 07 120590(V) =08 120590(119908) = 1 and 120590(119909) = 06 Define a fuzzy set 120583 onE such that 120583(119906 V) = 120583(119906 119908) = 07 120583(119906 119909) = 120583(V 119909) =120583(119908 119909) = 06 and 120583(V 119908) = 08 Then 120583(119906 V) = 120590(119906) and 120590(V)for all 119906 V isin 119881 Thus 119866 = (119881 120590 120583) is a complete fuzzygraph Now119866lowast = (119881lowast 119864lowast) where 119881lowast = 119906 isin 119881120590(119906) gt0 = 119906 V 119908 119909 and 119864lowast = (119906 V) 119906 V isin 119881120583(119906 V) gt0 = (119906 V) (119906 119908) (119906 119909) (V 119909) (119908 119909) (V 119908) This showsthat each vertex in 119866lowast joins each of the other vertices in 119866lowastexactly by one edge Therefore 119866lowast is a complete crisp graph1198704

Remark 46 For complete fuzzy graph 119866 with n vertices byLemma 44 119866lowast = 119870119899 where 119870119899 is a complete crisp graph

The following is an important theorem for computingthe fuzzy chromatic polynomial of a complete fuzzy graphexplicitly

Theorem 47 Let 119866 be a complete fuzzy graph with n verticesIf 120572 isin 119868 = 119871 cup 0 and 120573 = min(119871) then

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 and 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1(10)

Proof Let 119866 = (119881 120590 120583) be a complete fuzzy graph with nvertices and let 119866lowast = (119881lowast 119864lowast) be the underlying crisp graphof 119866 where 119881lowast = 119906 isin 119881 | 120590(119906) gt 0 and 119864lowast = (119906 V) 119906 V isin119881 | 120583(119906 V) gt 0 For 120572 isin I = L cup 0 where L is a level setof 119866 now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572and 119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen by Theorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (11)

and since 119866 is complete fuzzy graph then by Remark 46 and(11) we get 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896)This completes theproof

Remark 48 For a complete fuzzy graph 119866 = (119881 120590 120583) withcrisp vertices (ie 120590(V) = 1 forallV isin 119881) and |119881| = 119899 119875119891120572 (119866 119896) =119875(119870119899 119896) for all 120572 isin I

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

Advances in Fuzzy Systems 9

The fuzzy chromatic polynomial of a complete fuzzygraph with |119871| = |119881| can be defined in terms of the chromaticpolynomial of complete crisp graph as follows

Definition 49 Let 119866 = (119881 120590 120583) be a complete fuzzy graphwith |119881| = 119899 vertices Let 119871 be a level set of 119866 and 120572 isin 119868 =1198711198800 Define 1205730 = min(119871) = 120573 1205731 = min(L1205730) 1205732 =min(L1205730 1205731) 120573119899minus1 = min(L1205730 1205731 120573119899minus2)

If |119871| = |119881| = 119899 119899 ge 1 then the fuzzy chromaticpolynomial of a complete fuzzy graph 119866 is defined as

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0 amp 1205730119875 (119870119899minus1 119896) 119894119891 120572 = 1205731119875 (119870119899minus2 119896) 119894119891 120572 = 1205732

119875 (1198701 119896) 119894119891 120572 = 120573119899minus1

(12)

where119870119899 119899 ge 1 is a complete crisp graph with n verticesNote that the advantage of Definition 49 is that the fuzzy

chromatic polynomial of a complete fuzzy graph with |119881| =|119871| can be obtainedwithout using the formal proceduresThissituation can be illustrated in the following example

Example 50 Consider the complete fuzzy graph 119866 in Fig-ure 7

In Figure 7 |119881| = 4 119868 = 119871cup0 where L = 06 07 08 1Therefore |119871| = 4 Also 1205730 = 06 1205731 = 07 1205732 = 08 and1205733 = 1

Since |119871| = |119881| = 4 then by Definition 49 the fuzzychromatic polynomial of the complete fuzzy graph 119866 is

119875119891120572 (119866 119896) =

119896 (119896 minus 1) (119896 minus 2) (119896 minus 3) 120572 = 0 amp 06

119896 (119896 minus 1) (119896 minus 2) 120572 = 07

119896 (119896 minus 1) 120572 = 08

119896 120572 = 1

(13)

A fuzzy graph is a cycle and a fuzzy cycle as defined in [9 32]

Definition 51 (see [9 32]) Let 119866 = (119881 120590 120583) be a fuzzy graphThen

(i) G is called a cycle if 119866lowast = (119881lowast 119864lowast) is a cycle

(ii) G is called a fuzzy cycle if119866lowast = (119881lowast 119864lowast) is a cycle and∄ unique (119909 119910) isin 119864lowast such that 120583(119909 119910) = and120583(119906 V) |(119906 V) isin 119864lowast

Remark 52 If a fuzzy graph 119866 is cycle with n vertices then119866lowast = 119862119899 where 119862119899 is a cycle crisp graph with n vertices

The following theorem is important for computing thefuzzy chromatic polynomial of a fuzzy graph which is cycle

08

06

06 0607

07Ｐ1(07) Ｐ2(08)

Ｐ3(1)Ｐ4(06)

Figure 7 A complete fuzzy graph 119866 with |119881| = |119871|

Theorem 53 Let119866 be a fuzzy graph with 119899 vertices en119866 isa cycle if and only if

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(14)

where 119870119899 and 119862119899 are a complete crisp graph and cycle crispgraph with n vertices respectively

Proof Let 119866 = (119881 120590 120583) be a fuzzy graph with n verticesSuppose 119866 be a cycle Since 119866 is a cycle 119866lowast is a cycle crispgraph For 120572 isin I = L cup 0 where L is a level set of 119866now 119866120572 = (119881120572 119864120572) where 119881120572 = 119906 isin 119881 | 120590(119906) ge 120572 and119864120572 = (119906 V) | 120583(119906 V) ge 120572 Let us prove the theorem byconsidering three cases

Case 1 Let 120572 = 0 and since 119866 is a fuzzy graph with n verticesthen byTheorem 39 we have 119875119891120572 (119866 119896) = 119875(119870119899 119896)

Case 2 Let 120572 = 120573 where 120573 = min(119871) since119866 is a fuzzy graphthen byTheorem 40 we have

119875119891120572 (119866 119896) = 119875 (119866lowast 119896) (15)

and since G is cycle with n vertices then by Remark 52 and(15) we get 119875119891120572 (119866 119896) = 119875(119862119899 119896)

Case 3 Let 120573 lt 120572 le 1 since 120572 is in I and 119866 is a fuzzy graphthen by Theorem 41 119875119891120572 (119866 119896) = 119875(119866120572 119896) Hence the resultholds

Conversely we shall prove it by contrapositive Suppose119866 is not a cycle Then by Definition 51 119866lowast is not a cycle Thisimplies that 119866lowast = 119862119899 Hence

119875119891120572 (119866 119896) =

119875 (119870119899 119896) 119894119891 120572 = 0

119875 (119862119899 119896) 119894119891 120572 = 120573

119875 (119866120572 119896) 119894119891 120573 lt 120572 le 1

(16)

Therefore if (14) is true then 119866 is a cycle This completes theproof

Corollary 54 Let119866 be a fuzzy graph If119866 is a fuzzy cycle with119899 vertices then (14) is true

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

10 Advances in Fuzzy Systems

08

02

06 06

x w

vu

Figure 8 A fuzzy graph 119866

Proof Let 119866 be a fuzzy cycle with 119899 vertices Since 119866 is fuzzycycle then its 119866lowast is cycle Then by Definition 51 119866 is a cycleThen byTheorem 53 the result is immediate

Remark 55 The converse of Corollary 54 is not true It can beseen from the following example

Example 56 Consider the fuzzy graph 119866 given in Figure 8

In Figure 8 clearly 119866lowast is a cycle So 119866 is cycle For 120572 isin119868 = 119871 cup 0where 119871 = 02 06 08 1 since G is cycle thenby (14) the fuzzy chromatic polynomial of 119866 is

119875119891120572 (119866 119896) =

119875 (1198704 119896) 119894119891 120572 = 0

119875 (1198624 119896) 119894119891 120572 = 02

119875 (119866120572 119896) 119894119891 02 lt 120572 le 1

(17)

But 119866 is not fuzzy cycle

Remark 57 In general the meaning of fuzzy chromaticpolynomial of a fuzzy graph depends on the sense of theindex 120572 It can be interpreted that for lower values of 120572more numbers of different k-colorings are obtained On theother hand for higher values of 120572 less number of differentk-colorings are obtained Therefore the fuzzy chromaticpolynomial sums up all this information so as to manage thefuzzy problem

5 Conclusions

In this paper the concept of fuzzy chromatic polynomial offuzzy graphwith crisp and fuzzy vertex sets is introducedThefuzzy chromatic polynomial of fuzzy graph is defined basedon 120572-cuts of the fuzzy graph Some properties of the fuzzychromatic polynomial of fuzzy graphs are investigated Thedegree of fuzzy chromatic polynomial of a fuzzy graph is lessthan or equal to the number of vertices in fuzzy graph whichis a significant difference from the crisp graph theory Furtherthe fuzzy chromatic polynomial of fuzzy graph with fuzzyvertex set will decreasewhen the value of120572will increase Alsomore results on fuzzy chromatic polynomial of fuzzy graphshave been proved The relation between the fuzzy chromaticpolynomial of fuzzy graph and its underlying crisp graph andthe relation between the fuzzy chromatic polynomial of fuzzygraph and complete crisp graph are established Finally thefuzzy chromatic polynomial for complete fuzzy graph andfuzzy cycle is studied Also the fuzzy chromatic polynomialof complete fuzzy graph is defined explicitly when |119881| = |119871|

We are working on fuzzy chromatic polynomial based ondifferent types of fuzzy coloring functions as an extension ofthis study

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Both authors contributed equally and significantly in writ-ing this article Both authors read and approved the finalmanuscript

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] C M Klein ldquoFuzzy shortest pathsrdquo Fuzzy Sets and Systems vol39 no 1 pp 27ndash41 1991

[3] K C Lin and M S Chern ldquoThe fuzzy shortest path problemand its most vital arcsrdquo Fuzzy Sets and Systems vol 58 no 3pp 343ndash353 1993

[4] S Okada and T Soper ldquoA shortest path problem on a networkwith fuzzy arc lengthsrdquo Fuzzy Sets and Systems vol 109 no 1pp 129ndash140 2000

[5] S M Nayeem andM Pal ldquoShortest path problem on a networkwith imprecise edge weightrdquo Fuzzy Optimization and DecisionMaking vol 4 no 4 pp 293ndash312 2005

[6] A Dey A Pal and T Pal ldquoInterval type 2 fuzzy set in fuzzyshortest path problemrdquoMathematics vol 4 no 4 pp 1ndash19 2016

[7] A Dey R Pradhan A Pal and T Pal ldquoA genetic algorithm forsolving fuzzy shortest path problems with interval type-2 fuzzyarc lengthsrdquoMalaysian Journal of Computer Science vol 31 no4 pp 255ndash270 2018

[8] A Kaufmann Introduction A La eorie Des Sous-EnsemblesFlous vol 1 Masson et Cie Paris 1973

[9] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and eir Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975

[10] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987

[11] K R Bhutani ldquoOn automorphisms of fuzzy graphsrdquo PatternRecognition Letters vol 9 no 3 pp 159ndash162 1989

[12] J N Mordeson and P S Nair ldquoCycles and cocycles of fuzzygraphsrdquo Information Sciences vol 90 no 1-4 pp 39ndash49 1996

[13] M S Sunitha and A Vijayakumar ldquoA characterization of fuzzytreesrdquo Information Sciences vol 113 no 3-4 pp 293ndash300 1999

[14] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002

[15] K R Bhutani and A Rosenfeld ldquoStrong arcs in fuzzy graphsrdquoInformation Sciences vol 152 pp 319ndash322 2003

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

Advances in Fuzzy Systems 11

[16] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009

[17] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[18] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011

[19] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica-Verlag Heidelberg Germany 2000

[20] M Ananthanarayanan and S Lavanya ldquoFuzzy graph coloringusing 120572 cutsrdquo International Journal of Engineering and AppliedSciences vol 4 no 10 pp 23ndash28 2014

[21] C Eslahchi and B N Onagh ldquoVertex-strength of fuzzy graphsrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 2006 Article ID 43614 9 pages 2006

[22] S Munoz M T Ortuno J Ramırez and J Yanez ldquoColoringfuzzy graphsrdquo Omega vol 33 no 3 pp 211ndash221 2005

[23] A Kishore and M S Sunitha ldquoChromatic number of fuzzygraphsrdquoAnnals of FuzzyMathematics and Informatics vol 7 no4 pp 543ndash551 2014

[24] A Dey and A Pal ldquoVertex coloring of a fuzzy graph using alphcutrdquo International Journal ofManagement vol 2 no 8 pp 340ndash352 2012

[25] A Dey and A Pal ldquoFuzzy graph coloring technique to classifythe accidental zone of a traffic controlrdquo Annals of Pure andApplied Mathematics vol 3 no 2 pp 169ndash178 2013

[26] A Dey R Pradhan A Pal and T Pal ldquoThe fuzzy robustgraph coloring problemrdquo in Proceedings of the 3rd InternationalConference on Frontiers of Intelligent Computingeory andApplications (FICTA) 2014 vol 327 pp 805ndash813 Springer 2015

[27] I Rosyida Widodo C R Indrati and K A Sugeng ldquoA newapproach for determining fuzzy chromatic number of fuzzygraphrdquo Journal of Intelligent and Fuzzy Systems Applications inEngineering and Technology vol 28 no 5 pp 2331ndash2341 2015

[28] S Samanta T Pramanik and M Pal ldquoFuzzy colouring of fuzzygraphsrdquo Afrika Matematika vol 27 no 1-2 pp 37ndash50 2016

[29] A Dey L H Son P K K Kumar G Selvachandran and SG Quek ldquoNew concepts on vertex and edge coloring of simplevague graphsrdquo Symmetry vol 10 no 9 pp 1ndash18 2018

[30] R A Brualdi Introductory Combinatorics Prentice Hall 2009[31] R J Wilson Introduction to Graph eory Pearson England

2010[32] S Mathew J N Mordeson and D S Malik ldquoFuzzy graphsrdquo in

Fuzzy Graph eory vol 363 pp 13ndash83 Springer InternationalPublishing AG 2018

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawiwwwhindawicom Volume 2018

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Computational Intelligence and Neuroscience

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018

Human-ComputerInteraction

Advances in

Hindawiwwwhindawicom Volume 2018

Scientic Programming

Submit your manuscripts atwwwhindawicom

Computer Games Technology

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Advances in

FuzzySystems

Hindawiwwwhindawicom

Volume 2018

International Journal of

ReconfigurableComputing

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

thinspArtificial Intelligence

Hindawiwwwhindawicom Volumethinsp2018

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Hindawiwwwhindawicom Volume 2018

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