Research ArticleOn Super Mean Labeling for Total Graph of Path and Cycle

Nur Inayah 1 I Wayan Sudarsana2 Selvy Musdalifah2 and Nurhasanah DaengMangesa2

1Mathematics Department Faculty of Sciences and Technology State Islamic University of Syarif Hidayatullah Jakarta Indonesia2Combinatorial and Applied Mathematics Research Group (CAMRG) Department of MathematicsFaculty of Mathematics and Natural Sciences Tadulako University Palu Indonesia

Correspondence should be addressed to Nur Inayah nurinayahuinjktacid

Received 27 April 2017 Revised 23 July 2017 Accepted 14 February 2018 Published 6 June 2018

Academic Editor Dalibor Froncek

Copyright copy 2018 Nur Inayah et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Let 119866(119881 119864) be a graph with the vertex set 119881 and the edge set 119864 respectively By a graph 119866 = (119881 119864) we mean a finite undirectedgraph with neither loops nor multiple edges The number of vertices of 119866 is called order of 119866 and it is denoted by 119901 Let 119866 be a(119901 119902) graph A super mean graph on 119866 is an injection 119891 119881 rarr 1 2 3 119901 + 119902 such that for each edge 119890 = 119906V in 119864 labeled by119891lowast(119890) = lceil(119891(119906) + 119891(V))2rceil the set 119891(119881) cup 119891lowast(119890) 119890 isin 119864 forms 1 2 3 119901 + 119902 A graph which admits super mean labeling iscalled super mean graph The total graph 119879(119866) of 119866 is the graph with the vertex set 119881 cup 119864 and two vertices are adjacent wheneverthey are either adjacent or incident in 119866 We have showed that graphs 119879(119875

119899) and 119879(119862

119899) are super mean where 119875

119899is a path on 119899

vertices and 119862119899is a cycle on 119899 vertices

1 Introduction and Preliminary Results

Let 119866(119881 119864) be a graph with the vertex set 119881 and the edgeset 119864 respectively By a graph 119866 = (119881 119864) we mean a finiteundirected graph with neither loops nor multiple edges Thenumber of vertices of 119866 is called order of 119866 and it is denotedby 119901 The number of edges of 119866 is called size of 119866 and it isdenoted by 119902 A (119901 119902) graph 119866 is a graph with 119901 vertices and119902 edges Terms and notations not defined here are used in thesense of Harary [1]

In 2003 Somasundaram and Ponraj [2] have introducedthe notion of mean labelings of graphs Let 119866 be a (119901 119902)graph A graph119866 is called amean graph if there is an injectivefunction 119891 from the vertices of G to 0 1 2 sdot sdot sdot 119902 such thatwhen each edge 119890 = 119906V is labeled with 119891lowast(119890) = (119891(119906) +119891(V) + 1)2 if 119891(119906) + 119891(V) is even and 119891lowast(119890) = (119891(119906) +119891(V) + 1)2 if 119891(119906) + 119891(V) is odd then the resulting edgelabels are distinct Furthermore the concept of super meanlabeling was introduced by Ponraj and Ramya [3] Let 119891 119881 rarr 1 2 3 sdot sdot sdot 119901+119902 be an injection on119866 For each edge 119890 =119906V and an integer 119898 ge 2 the induced Smarandachely edge119898minus119897119886119887119890119897119894119899119892 119891lowast is defined by119891lowast(119890 = 119906V) = lceil(119891(119906)+119891(V))119898rceilThen 119891 is called a Smarandachely super119898 minus 119898119890119886119899 labeling if119891(119881) cup 119891lowast(119890) 119890 isin 119864 = 1 2 3 sdot sdot sdot 119901 + 119902 A graph that

admits a Smarandachely super mean 119898 minus 119897119886119887119890119897119894119899119892 is calledSmarandachely super119898minus119898119890119886119899 graph Particularly if119898 = 2we know that

119891lowast (119890 = 119906V)

= 119891 (119906) + 119891 (V)2 if119891 (119906) + 119891 (V) is even119891 (119906) + 119891 (V) + 12 if119891 (119906) + 119891 (V) is odd

(1)

Such a labeling 119891 is called a super mean labeling of 119866 if119891(119881) cup 119891lowast(119890) 119890 isin 119864 = 1 2 3 sdot sdot sdot 119901 + 119902 A graph thatadmits a super mean labeling is called a super mean graphFurther discussions of mean and super mean labelings forsome families of graph are provided in [4ndash10] andGallian [11]

The total graph 119879(119866) of 119866 is the graph with the vertexset 119881 cup 119864 and two vertices are adjacent whenever they areeither adjacent or incident in 119866 For instance when 119866 = 119875

119899

total graph of path 119879(119875119899) is provided in Figure 1 Since the

problem on super mean labeling for total graph of path andcycle are still open the new our contributions are stated inthe following sections

HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2018 Article ID 9250424 5 pageshttpsdoiorg10115520189250424

2 International Journal of Mathematics and Mathematical Sciences

Ｏ1 Ｏ2 Ｏ3ＯＨ-2 ＯＨ-1

Ｐ1 Ｐ2 Ｐ3 ＰＨ-2 ＰＨ-1 ＰＨＨ-21 2 Ｈ-1

1

2

Ｈ-2

Ｈ-1

Ｈ-1

Ｈ-2

Ｈ-2

3

2

2

1

1

Figure 1 The total graph of path on n vertices

18 20

531

22

7 9

24

2 4 6 8

10 11 12 13 14 15 16 17

19 21 23

Figure 2 The super mean labeling for total graph of path on 5 vertices 119879(1198755)

2 On Super Mean Labeling for TotalGraph of Path

The theorem proposed in this section deals with the supermean labeling for total graph of path on 119899 vertices 119879(119875

119899)

Theorem 1 The total graph of path on n vertices 119879(119875119899) is a

super mean graph for all 119899 ge 3Proof Let 119881(119879(119875

119899)) = V

119894 1 le 119894 le 119899 cup 119906

119894 1 le 119894 le 119899 minus 1

and 119864(119879(119875119899)) = 119890

119894 1198901015840119894 11989010158401015840119894

1 le 119894 le 119899minus1cup 119890119898119894

1 le 119894 le 119899minus2with 119890

119894= V119894V119894+1

1198901015840119894

= V119894119906119894 11989010158401015840119894

= V119894+1

119906119894for 1 le 119894 le 119899ndash1

and 119890101584010158401015840119894

= 119906119894119906119894+1

for 1 le 119894 le 119899ndash2 Immediately we have thatthe cardinality of the vertex set and the edge set of 119879(119875

119899) are119901 = 2119899 minus 1 and 119902 = 4119899 minus 5 respectively and so 119901+ 119902 = 6119899 minus 6

Define an injection119891 119881(119879(119875119899)) rarr 1 2 6119899minus6 for 119899 ge 3

as follows 119891(V119894) = 2119894 minus 1 for 119894 = 1 2 119899 119891(119906

119894) = 2119894 + 4119899minus 4

for 119894 = 1 2 119899 minus 1And so we have

119891lowast (119890119894) = 2119894 for 119894 = 1 2 119899 minus 1

119891lowast (1198901015840119894) = 2119899 + 2119894 minus 2 for 119894 = 1 2 119899 minus 1

119891lowast (11989010158401015840119894) = 2119899 + 2119894 minus 1 for 119894 = 1 2 119899 minus 1

119891lowast (119890101584010158401015840119894

) = 4119899 + 2119894 minus 3 for 119894 = 1 2 119899 minus 2(2)

Next we consider the following sets

1198601= 119891 (V

119894) = 2119894 minus 1 119894 = 1 2 119899

1198602= 119891 (119906

119894) = 4119899 + 2119894 minus 4 119894 = 1 2 119899 minus 1

1198603= 119891lowast (119890

119894) = 2119894 119894 = 1 2 119899 minus 1

1198604= 119891lowast (1198901015840

119894) = 2119899 + 2119894 minus 2 119894 = 1 2 119899 minus 1

1198605= 119891lowast (11989010158401015840

119894) = 2119899 + 2119894 minus 1 119894 = 1 2 119899 minus 1

1198606= 119891lowast (119890101584010158401015840

119894) = 4119899 + 2119894 minus 3 119894 = 1 2 119899 minus 2

(3)

Ｏ1

Ｏ2

Ｏ3

ＯＨ

ＯＨ-1Ｐ1

Ｐ2

Ｐ3

ＰＨ-1

ＰＨ

1

2

3

Ｈ

1

Ｈ

2

Ｈ-1

Ｈ-1

Ｈ-1

Ｈ

2

2

1

Ｈ

1

3

Figure 3 The total graph of cycle on n vertices 119879(119862119899)

It can be verified that 119891(119881(119879(119875119899))) cup 119891lowast(119864(119879(119875

119899))) =⋃6

119894=1119860119894= 1 2 3 6119899minus6 and so119891 is a supermean labeling

of 119879(119875119899) Hence 119879(119875

119899) is a super mean graph For a simple

example the super mean labeling for total graph of path onfive vertices is provided in Figure 2

3 On Super Mean Labeling for TotalGraph of Cycle

The theorem proposed in this section deals with the supermean labeling for total graph of cycle on 119899 vertices 119879(119862

119899)

For illustration total graph of cycle on 119899 vertices is providedin Figure 3

Theorem 2 The total graph of cycle on n vertices 119879(119862119899) is a

super mean graph if either 119899 is odd and 119899 ge 3 or n is even and119899 ge 6

International Journal of Mathematics and Mathematical Sciences 3

Proof Let 119881(119879(119862119899)) = V

119894 119906119894 1 le 119894 le 119899 and 119864(119879(119862

119899)) =119890

119894 1198901015840119894 11989010158401015840119894 119890101584010158401015840119894

1 le 119894 le 119899 where119890119894= V119894V119894+1 1 le 119894 le 119899 minus 1

V1V119894 119894 = 119899

1198901015840119894= V119894119906119894

for 1 le 119894 le 11989911989010158401015840119894

= V119894+1119906119894 1 le 119894 le 119899 minus 1V1119906119894 119894 = 119899

119890101584010158401015840119894

= 119906119894119906119894+1

1 le 119894 le 119899 minus 11199061119906119894 119894 = 119899

(4)

Immediately we have that the cardinality of the vertex set andthe edge set of 119879(119862

119899) are 119901 = 2119899 and 119902 = 4119899 respectively and

so 119901 + 119902 = 6119899Define an injection 119891 119881(119879(119862

119899)) rarr 1 2 6119899 for odd119899 ge 3 as follows

119891 (V119894) =

2119894 minus 1 119894 = 1 2 3 lceil1198992rceil 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

119891 (119906119894) =

4119899 + 2119894 minus 1 119894 = 1 2 3 lceil1198992rceil 4119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

(5)

And so we have

119891lowast (119890119894) =

2119894 119894 = 1 2 3 lceil1198992rceil minus 1119899 + 2 119894 = lceil1198992rceil 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899 minus 1119899 + 1 119894 = 119899

119891 (1198901015840119894) =

2119899 + 2119894 minus 1 119894 = 1 2 3 lceil1198992rceil 2119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

119891lowast (11989010158401015840119894)

=

2119899 + 2119894 119894 = 1 2 3 lceil1198992rceil minus 13119899 + 2 119894 = lceil1198992rceil 2119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899 minus 13119899 + 1 119894 = 119899

119891lowast (119890101584010158401015840119894

)

=

4119899 + 2119894 119894 = 1 2 3 lceil1198992rceil minus 15119899 + 2 119894 = lceil1198992rceil 4119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899 minus 15119899 + 1 119894 = 119899

(6)

Next we consider the following sets

1198601= 119891 (V

119894) = 2119894 minus 1 119894 = 1 2 lceil1198992rceil

1198602= 119891 (V

119894) = 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

1198603= 119891 (119906

119894) = 4119899 + 2119894 minus 1 119894 = 1 2 lceil1198992rceil

1198604= 119891 (119906

119894) = 4119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

1198605= 119891lowast (119890

119894) = 2119894 119894 = 1 2 lceil1198992rceil minus 1

1198606= 119891lowast (119890

119894) = 119899 + 2 119894 = lceil1198992rceil

1198607= 119891lowast (119890

119894) = 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

minus 1 1198608= 119891 (119890

119894) = 119899 + 1 119894 = 119899

1198609= 119891lowast (1198901015840

119894) = 2119899 + 2119894 minus 1 119894 = 1 2 lceil1198992rceil

11986010

= 119891lowast (1198901015840119894) = 2119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil

+ 2 119899 11986011

= 119891lowast (11989010158401015840119894) = 2119899 + 2119894 119894 = 1 2 lceil1198992rceil minus 1

11986012

= 119891lowast (11989010158401015840119894) = 3119899 + 2 119894 = lceil1198992rceil

11986013

= 119891lowast (11989010158401015840119894) = 2119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil

+ 2 119899 minus 1 11986014

= 119891lowast (11989010158401015840119894) = 3119899 + 1 119894 = 119899

11986015

= 119891lowast (119890101584010158401015840119894

) = 4119899 + 2119894 119894 = 1 2 lceil1198992rceil minus 1 11986016

= 119891lowast (119890101584010158401015840119894

) = 5119899 + 2 119894 = lceil1198992rceil 11986017

= 119891lowast (119890101584010158401015840119894

) = 4119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil+ 2 119899 minus 1

11986018

= 119891lowast (119890101584010158401015840119894

) = 5119899 + 1 119894 = 119899

(7)

It can be verified that 119891(119881(119879(119862119899))) cup 119891lowast(119864(119879(119862

119899))) =⋃18

119894=1119860119894= 1 2 3 6119899 and so 119891 is a super mean labeling

of 119879(119862119899) Hence 119879(119862

119899) is a super mean graph for odd 119899 ge 3

4 International Journal of Mathematics and Mathematical Sciences

Now define an injection 1198911 119881(119879(119862

119899)) rarr 1 2 3 6119899

for even 119899 ge 6 as follows

1198911(V119894) =

1 119894 = 13119894 minus 3 119894 = 2 34119894 minus 7 119894 = 4 5 6 1198992 + 14119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 17 119894 = 119899

1198911(119906119894) =

4119899 + 1 119894 = 14119899 + 3119894 minus 3 119894 = 2 34119899 + 4119894 minus 7 119894 = 4 5 6 1198992 + 18119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 14119899 + 7 119894 = 119899

(8)

and so we have

119891lowast1(119890119894) =

2 119894 = 13119894 minus 1 119894 = 2 34119894 minus 5 119894 = 4 5 6 1198992 + 14119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 14 119894 = 119899

119891lowast1(1198901015840119894) =

2119899 + 1 119894 = 12119899 + 3119894 minus 3 119894 = 2 32119899 + 4119894 minus 7 119894 = 4 5 6 1198992 + 16119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 + 7 119894 = 119899

119891lowast1(11989010158401015840119894) =

2119899 + 2 119894 = 12119899 + 3119894 minus 1 119894 = 2 32119899 + 4119894 minus 5 119894 = 4 5 6 1198992 + 16119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 + 4 119894 = 119899

119891lowast1(119890101584010158401015840119894

) =

4119899 + 2 119894 = 14119899 + 3119894 minus 1 119894 = 2 34119899 + 4119894 minus 5 119894 = 4 5 6 1198992 + 18119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 14119899 + 4 119894 = 119899

(9)

It can be verified that 1198911(119881(119879(119862

119899))) cup 119891lowast

1(119864(119879(119862

119899))) =1 2 3 6119899 and so119891

1is a super mean labeling of 119879(119862

119899)

Hence 119879(119862119899) is a super mean graph for even 119899 ge 6 For

45

48

44

38

35

33

41

39

1

3

6

9

7

12

16

13

2

5

811

15

14

10

4

17

18

19

21

22

2425 27

29

31

32

28

2623

34

30

20

37

40 43

47

46

4236

Figure 4 The super mean labeling for total graph of cycle on 8vertices 119879(119862

8)

illustration a super mean labeling for total graph of cycle on8vertices is provided in Figure 4

4 The Duality of Super Mean Labeling

Let 119866 be a (119901 119902) graph Given any Smarandachely super 2 minus119898119890119886119899 labeling 120582 on graph 119866 the labeling 1205821defined by

1205821 (119909) = 119901 + 119902 + 1 minus 120582 (119909) for any vertex 119909

120582lowast1(119909119910) = 119901 + 119902 + 1 minus 120582lowast (119909119910) for any edges 119909119910 (10)

is also a Smarandachely super 2 minus 119898119890119886119899 labeling of 119866For the proof since 120582 is an injection then it is follows that1205821is also an injection on 119866 Hence it can be verified that the

set 1205821(119881)cup120582lowast1(119890) 119890 isin 119864 forms 1 2 3 sdot sdot sdot 119901+119902 and so the

injection 1205821is also a Smarandachely super 2 minus 119898119890119886119899 labeling

on graph119866 Furthermore we call that the labeling 1205821is a dual

super mean labeling of 120582By using the duality property aboveTheorems 1 and 2 we

have the following corollary

Corollary 3 Let 119879(119875119899) and 119879(119862

119899) be the total graph of path

and cycle with 119899 vertices respectively(i) For all 119899 ge 3 if 120582(V

119894) = 6119899 minus 2119894 minus 4 for 1 le 119894 le 119899 and120582(119906

119894) = 2119899 minus 2119894 minus 1 for 1 le 119894 le 119899 minus 1 then 120582 is a super

mean labeling for 119879(119875119899)

(ii) For odd 119899 ge 3 if120582 (V119894) =

6119899 minus 2119894 + 2 119894 = 1 2 3 lceil1198992rceil 6119899 minus 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

120582 (119906119894) =

2119899 minus 2119894 + 2 119894 = 1 2 3 lceil1198992rceil 2119899 minus 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

(11)

then 120582 is a super mean labeling for 119879(119862119899)

International Journal of Mathematics and Mathematical Sciences 5

(iii) For even 119899 ge 6 if

1205821(V119894) =

6119899 119894 = 16119899 minus 3119894 + 4 119894 = 2 36119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 + 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 16119899 minus 6 119894 = 119899

1205821(119906119894) =

2119899 119894 = 12119899 minus 3119894 + 4 119894 = 2 32119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 minus 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 minus 6 119894 = 119899

(12)

then 1205821is a super mean labeling for 119879(119862

119899)

5 Summary and Remarks

Here we propose new results corresponding to super meanlabeling for total graph of path and cycleThiswork is an effortto relate Smarandachely super119898minus119898119890119886119899 labeling and its dualfor 119898 ge 2 All results reported here are in total graph of pathand cycle119879(119875

119899) and119879(119862

119899) In future it is not only possible to

investigate some more results corresponding to other graphfamilies but also Smarandachely super m-mean labeling ingeneral as well

Disclosure

An earlier version of this paper was presented as an abstractat Distace in Graph 2016

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to express their very great apprecia-tion to Dr I Wayan Sudarsana Mrs Selvy Musdalifah andMrs Nurhasanah DaengMangesa for their valuable and con-structive suggestion during the conducting of this researchwork Their willingness to give their time so generously hasbeen very much appreciated

References

[1] F Harary ldquoRecent results on generalized Ramsey theory forgraphsrdquo in Graph Theory and Applications vol 303 of LectureNotes in Mathematics pp 125ndash138 Springer Berlin HeidelbergBerlin Heidelberg 1972

[2] S Somasundaram and R Ponraj ldquoMean labelings of graphsrdquoNational Academy of Science Letters vol 26 no 7-8 pp 210ndash213 2003

[3] R Ponraj and D Ramya ldquoOn super mean graphs of orderrdquoBulletin of Pure amp Applied Sciences vol 25 no 1 pp 143ndash1482006

[4] S Somasundaram and R Ponraj ldquoNon-existence of meanlabeling for a wheelrdquo Bulletin of Pure amp Applied Sciences-Mathematics and Statistics vol 22E pp 103ndash111 2003

[5] S Somasundaram and R Ponraj ldquoSome results on meangraphsrdquo Pure and Applied Mathematics Journal vol 58 pp 29ndash35 2003

[6] P Jeyanthi and D Ramya ldquoSuper mean labeling of some classesof graphsrdquo International Journal of Mathematical Combina-torics vol 1 pp 83ndash91 2012

[7] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meangraphsrdquo AKCE International Journal of Graphs and Combina-torics vol 6 no 1 pp 103ndash112 2009

[8] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meanlabeling of some graphsrdquo SUT Journal of Mathematics vol 46no 1 pp 53ndash66 2010

[9] P Jeyanthi and D Ramya ldquoSuper mean graphsrdquoUtilitas Mathe-matica vol 96 pp 101ndash109 2015

[10] D Ramya R Ponraj and P Jeyanthi ldquoSuper mean labeling ofgraphsrdquo Ars Combinatoria vol 112 pp 65ndash72 2013

[11] J A Gallian ldquoA dynamic survey of graph labellingsrdquo TheElectronic Journal of Combinatorics 2016

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International Journal of Mathematics and Mathematical Sciences 3

Proof Let 119881(119879(119862119899)) = V

119894 119906119894 1 le 119894 le 119899 and 119864(119879(119862

119899)) =119890

119894 1198901015840119894 11989010158401015840119894 119890101584010158401015840119894

1 le 119894 le 119899 where119890119894= V119894V119894+1 1 le 119894 le 119899 minus 1

V1V119894 119894 = 119899

1198901015840119894= V119894119906119894

for 1 le 119894 le 11989911989010158401015840119894

= V119894+1119906119894 1 le 119894 le 119899 minus 1V1119906119894 119894 = 119899

119890101584010158401015840119894

= 119906119894119906119894+1

1 le 119894 le 119899 minus 11199061119906119894 119894 = 119899

(4)

Immediately we have that the cardinality of the vertex set andthe edge set of 119879(119862

119899) are 119901 = 2119899 and 119902 = 4119899 respectively and

so 119901 + 119902 = 6119899Define an injection 119891 119881(119879(119862

119899)) rarr 1 2 6119899 for odd119899 ge 3 as follows

119891 (V119894) =

2119894 minus 1 119894 = 1 2 3 lceil1198992rceil 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

119891 (119906119894) =

4119899 + 2119894 minus 1 119894 = 1 2 3 lceil1198992rceil 4119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

(5)

And so we have

119891lowast (119890119894) =

2119894 119894 = 1 2 3 lceil1198992rceil minus 1119899 + 2 119894 = lceil1198992rceil 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899 minus 1119899 + 1 119894 = 119899

119891 (1198901015840119894) =

2119899 + 2119894 minus 1 119894 = 1 2 3 lceil1198992rceil 2119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

119891lowast (11989010158401015840119894)

=

2119899 + 2119894 119894 = 1 2 3 lceil1198992rceil minus 13119899 + 2 119894 = lceil1198992rceil 2119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899 minus 13119899 + 1 119894 = 119899

119891lowast (119890101584010158401015840119894

)

=

4119899 + 2119894 119894 = 1 2 3 lceil1198992rceil minus 15119899 + 2 119894 = lceil1198992rceil 4119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899 minus 15119899 + 1 119894 = 119899

(6)

Next we consider the following sets

1198601= 119891 (V

119894) = 2119894 minus 1 119894 = 1 2 lceil1198992rceil

1198602= 119891 (V

119894) = 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

1198603= 119891 (119906

119894) = 4119899 + 2119894 minus 1 119894 = 1 2 lceil1198992rceil

1198604= 119891 (119906

119894) = 4119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

1198605= 119891lowast (119890

119894) = 2119894 119894 = 1 2 lceil1198992rceil minus 1

1198606= 119891lowast (119890

119894) = 119899 + 2 119894 = lceil1198992rceil

1198607= 119891lowast (119890

119894) = 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

minus 1 1198608= 119891 (119890

119894) = 119899 + 1 119894 = 119899

1198609= 119891lowast (1198901015840

119894) = 2119899 + 2119894 minus 1 119894 = 1 2 lceil1198992rceil

11986010

= 119891lowast (1198901015840119894) = 2119899 + 2119894 119894 = lceil1198992rceil + 1 lceil1198992rceil

+ 2 119899 11986011

= 119891lowast (11989010158401015840119894) = 2119899 + 2119894 119894 = 1 2 lceil1198992rceil minus 1

11986012

= 119891lowast (11989010158401015840119894) = 3119899 + 2 119894 = lceil1198992rceil

11986013

= 119891lowast (11989010158401015840119894) = 2119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil

+ 2 119899 minus 1 11986014

= 119891lowast (11989010158401015840119894) = 3119899 + 1 119894 = 119899

11986015

= 119891lowast (119890101584010158401015840119894

) = 4119899 + 2119894 119894 = 1 2 lceil1198992rceil minus 1 11986016

= 119891lowast (119890101584010158401015840119894

) = 5119899 + 2 119894 = lceil1198992rceil 11986017

= 119891lowast (119890101584010158401015840119894

) = 4119899 + 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil+ 2 119899 minus 1

11986018

= 119891lowast (119890101584010158401015840119894

) = 5119899 + 1 119894 = 119899

(7)

It can be verified that 119891(119881(119879(119862119899))) cup 119891lowast(119864(119879(119862

119899))) =⋃18

119894=1119860119894= 1 2 3 6119899 and so 119891 is a super mean labeling

of 119879(119862119899) Hence 119879(119862

119899) is a super mean graph for odd 119899 ge 3

4 International Journal of Mathematics and Mathematical Sciences

Now define an injection 1198911 119881(119879(119862

119899)) rarr 1 2 3 6119899

for even 119899 ge 6 as follows

1198911(V119894) =

1 119894 = 13119894 minus 3 119894 = 2 34119894 minus 7 119894 = 4 5 6 1198992 + 14119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 17 119894 = 119899

1198911(119906119894) =

4119899 + 1 119894 = 14119899 + 3119894 minus 3 119894 = 2 34119899 + 4119894 minus 7 119894 = 4 5 6 1198992 + 18119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 14119899 + 7 119894 = 119899

(8)

and so we have

119891lowast1(119890119894) =

2 119894 = 13119894 minus 1 119894 = 2 34119894 minus 5 119894 = 4 5 6 1198992 + 14119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 14 119894 = 119899

119891lowast1(1198901015840119894) =

2119899 + 1 119894 = 12119899 + 3119894 minus 3 119894 = 2 32119899 + 4119894 minus 7 119894 = 4 5 6 1198992 + 16119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 + 7 119894 = 119899

119891lowast1(11989010158401015840119894) =

2119899 + 2 119894 = 12119899 + 3119894 minus 1 119894 = 2 32119899 + 4119894 minus 5 119894 = 4 5 6 1198992 + 16119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 + 4 119894 = 119899

119891lowast1(119890101584010158401015840119894

) =

4119899 + 2 119894 = 14119899 + 3119894 minus 1 119894 = 2 34119899 + 4119894 minus 5 119894 = 4 5 6 1198992 + 18119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 14119899 + 4 119894 = 119899

(9)

It can be verified that 1198911(119881(119879(119862

119899))) cup 119891lowast

1(119864(119879(119862

119899))) =1 2 3 6119899 and so119891

1is a super mean labeling of 119879(119862

119899)

Hence 119879(119862119899) is a super mean graph for even 119899 ge 6 For

45

48

44

38

35

33

41

39

1

3

6

9

7

12

16

13

2

5

811

15

14

10

4

17

18

19

21

22

2425 27

29

31

32

28

2623

34

30

20

37

40 43

47

46

4236

Figure 4 The super mean labeling for total graph of cycle on 8vertices 119879(119862

8)

illustration a super mean labeling for total graph of cycle on8vertices is provided in Figure 4

4 The Duality of Super Mean Labeling

Let 119866 be a (119901 119902) graph Given any Smarandachely super 2 minus119898119890119886119899 labeling 120582 on graph 119866 the labeling 1205821defined by

1205821 (119909) = 119901 + 119902 + 1 minus 120582 (119909) for any vertex 119909

120582lowast1(119909119910) = 119901 + 119902 + 1 minus 120582lowast (119909119910) for any edges 119909119910 (10)

is also a Smarandachely super 2 minus 119898119890119886119899 labeling of 119866For the proof since 120582 is an injection then it is follows that1205821is also an injection on 119866 Hence it can be verified that the

set 1205821(119881)cup120582lowast1(119890) 119890 isin 119864 forms 1 2 3 sdot sdot sdot 119901+119902 and so the

injection 1205821is also a Smarandachely super 2 minus 119898119890119886119899 labeling

on graph119866 Furthermore we call that the labeling 1205821is a dual

super mean labeling of 120582By using the duality property aboveTheorems 1 and 2 we

have the following corollary

Corollary 3 Let 119879(119875119899) and 119879(119862

119899) be the total graph of path

and cycle with 119899 vertices respectively(i) For all 119899 ge 3 if 120582(V

119894) = 6119899 minus 2119894 minus 4 for 1 le 119894 le 119899 and120582(119906

119894) = 2119899 minus 2119894 minus 1 for 1 le 119894 le 119899 minus 1 then 120582 is a super

mean labeling for 119879(119875119899)

(ii) For odd 119899 ge 3 if120582 (V119894) =

6119899 minus 2119894 + 2 119894 = 1 2 3 lceil1198992rceil 6119899 minus 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

120582 (119906119894) =

2119899 minus 2119894 + 2 119894 = 1 2 3 lceil1198992rceil 2119899 minus 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

(11)

then 120582 is a super mean labeling for 119879(119862119899)

International Journal of Mathematics and Mathematical Sciences 5

(iii) For even 119899 ge 6 if

1205821(V119894) =

6119899 119894 = 16119899 minus 3119894 + 4 119894 = 2 36119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 + 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 16119899 minus 6 119894 = 119899

1205821(119906119894) =

2119899 119894 = 12119899 minus 3119894 + 4 119894 = 2 32119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 minus 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 minus 6 119894 = 119899

(12)

then 1205821is a super mean labeling for 119879(119862

119899)

5 Summary and Remarks

Here we propose new results corresponding to super meanlabeling for total graph of path and cycleThiswork is an effortto relate Smarandachely super119898minus119898119890119886119899 labeling and its dualfor 119898 ge 2 All results reported here are in total graph of pathand cycle119879(119875

119899) and119879(119862

119899) In future it is not only possible to

investigate some more results corresponding to other graphfamilies but also Smarandachely super m-mean labeling ingeneral as well

Disclosure

An earlier version of this paper was presented as an abstractat Distace in Graph 2016

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to express their very great apprecia-tion to Dr I Wayan Sudarsana Mrs Selvy Musdalifah andMrs Nurhasanah DaengMangesa for their valuable and con-structive suggestion during the conducting of this researchwork Their willingness to give their time so generously hasbeen very much appreciated

References

[1] F Harary ldquoRecent results on generalized Ramsey theory forgraphsrdquo in Graph Theory and Applications vol 303 of LectureNotes in Mathematics pp 125ndash138 Springer Berlin HeidelbergBerlin Heidelberg 1972

[2] S Somasundaram and R Ponraj ldquoMean labelings of graphsrdquoNational Academy of Science Letters vol 26 no 7-8 pp 210ndash213 2003

[3] R Ponraj and D Ramya ldquoOn super mean graphs of orderrdquoBulletin of Pure amp Applied Sciences vol 25 no 1 pp 143ndash1482006

[4] S Somasundaram and R Ponraj ldquoNon-existence of meanlabeling for a wheelrdquo Bulletin of Pure amp Applied Sciences-Mathematics and Statistics vol 22E pp 103ndash111 2003

[5] S Somasundaram and R Ponraj ldquoSome results on meangraphsrdquo Pure and Applied Mathematics Journal vol 58 pp 29ndash35 2003

[6] P Jeyanthi and D Ramya ldquoSuper mean labeling of some classesof graphsrdquo International Journal of Mathematical Combina-torics vol 1 pp 83ndash91 2012

[7] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meangraphsrdquo AKCE International Journal of Graphs and Combina-torics vol 6 no 1 pp 103ndash112 2009

[8] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meanlabeling of some graphsrdquo SUT Journal of Mathematics vol 46no 1 pp 53ndash66 2010

[9] P Jeyanthi and D Ramya ldquoSuper mean graphsrdquoUtilitas Mathe-matica vol 96 pp 101ndash109 2015

[10] D Ramya R Ponraj and P Jeyanthi ldquoSuper mean labeling ofgraphsrdquo Ars Combinatoria vol 112 pp 65ndash72 2013

[11] J A Gallian ldquoA dynamic survey of graph labellingsrdquo TheElectronic Journal of Combinatorics 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 International Journal of Mathematics and Mathematical Sciences

Now define an injection 1198911 119881(119879(119862

119899)) rarr 1 2 3 6119899

for even 119899 ge 6 as follows

1198911(V119894) =

1 119894 = 13119894 minus 3 119894 = 2 34119894 minus 7 119894 = 4 5 6 1198992 + 14119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 17 119894 = 119899

1198911(119906119894) =

4119899 + 1 119894 = 14119899 + 3119894 minus 3 119894 = 2 34119899 + 4119894 minus 7 119894 = 4 5 6 1198992 + 18119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 14119899 + 7 119894 = 119899

(8)

and so we have

119891lowast1(119890119894) =

2 119894 = 13119894 minus 1 119894 = 2 34119894 minus 5 119894 = 4 5 6 1198992 + 14119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 14 119894 = 119899

119891lowast1(1198901015840119894) =

2119899 + 1 119894 = 12119899 + 3119894 minus 3 119894 = 2 32119899 + 4119894 minus 7 119894 = 4 5 6 1198992 + 16119899 minus 4119894 + 8 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 + 7 119894 = 119899

119891lowast1(11989010158401015840119894) =

2119899 + 2 119894 = 12119899 + 3119894 minus 1 119894 = 2 32119899 + 4119894 minus 5 119894 = 4 5 6 1198992 + 16119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 + 4 119894 = 119899

119891lowast1(119890101584010158401015840119894

) =

4119899 + 2 119894 = 14119899 + 3119894 minus 1 119894 = 2 34119899 + 4119894 minus 5 119894 = 4 5 6 1198992 + 18119899 minus 4119894 + 6 119894 = 1198992 + 2 1198993 + 3 119899 minus 14119899 + 4 119894 = 119899

(9)

It can be verified that 1198911(119881(119879(119862

119899))) cup 119891lowast

1(119864(119879(119862

119899))) =1 2 3 6119899 and so119891

1is a super mean labeling of 119879(119862

119899)

Hence 119879(119862119899) is a super mean graph for even 119899 ge 6 For

45

48

44

38

35

33

41

39

1

3

6

9

7

12

16

13

2

5

811

15

14

10

4

17

18

19

21

22

2425 27

29

31

32

28

2623

34

30

20

37

40 43

47

46

4236

Figure 4 The super mean labeling for total graph of cycle on 8vertices 119879(119862

8)

illustration a super mean labeling for total graph of cycle on8vertices is provided in Figure 4

4 The Duality of Super Mean Labeling

Let 119866 be a (119901 119902) graph Given any Smarandachely super 2 minus119898119890119886119899 labeling 120582 on graph 119866 the labeling 1205821defined by

1205821 (119909) = 119901 + 119902 + 1 minus 120582 (119909) for any vertex 119909

120582lowast1(119909119910) = 119901 + 119902 + 1 minus 120582lowast (119909119910) for any edges 119909119910 (10)

is also a Smarandachely super 2 minus 119898119890119886119899 labeling of 119866For the proof since 120582 is an injection then it is follows that1205821is also an injection on 119866 Hence it can be verified that the

set 1205821(119881)cup120582lowast1(119890) 119890 isin 119864 forms 1 2 3 sdot sdot sdot 119901+119902 and so the

injection 1205821is also a Smarandachely super 2 minus 119898119890119886119899 labeling

on graph119866 Furthermore we call that the labeling 1205821is a dual

super mean labeling of 120582By using the duality property aboveTheorems 1 and 2 we

have the following corollary

Corollary 3 Let 119879(119875119899) and 119879(119862

119899) be the total graph of path

and cycle with 119899 vertices respectively(i) For all 119899 ge 3 if 120582(V

119894) = 6119899 minus 2119894 minus 4 for 1 le 119894 le 119899 and120582(119906

119894) = 2119899 minus 2119894 minus 1 for 1 le 119894 le 119899 minus 1 then 120582 is a super

mean labeling for 119879(119875119899)

(ii) For odd 119899 ge 3 if120582 (V119894) =

6119899 minus 2119894 + 2 119894 = 1 2 3 lceil1198992rceil 6119899 minus 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

120582 (119906119894) =

2119899 minus 2119894 + 2 119894 = 1 2 3 lceil1198992rceil 2119899 minus 2119894 + 1 119894 = lceil1198992rceil + 1 lceil1198992rceil + 2 119899

(11)

then 120582 is a super mean labeling for 119879(119862119899)

International Journal of Mathematics and Mathematical Sciences 5

(iii) For even 119899 ge 6 if

1205821(V119894) =

6119899 119894 = 16119899 minus 3119894 + 4 119894 = 2 36119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 + 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 16119899 minus 6 119894 = 119899

1205821(119906119894) =

2119899 119894 = 12119899 minus 3119894 + 4 119894 = 2 32119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 minus 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 minus 6 119894 = 119899

(12)

then 1205821is a super mean labeling for 119879(119862

119899)

5 Summary and Remarks

Here we propose new results corresponding to super meanlabeling for total graph of path and cycleThiswork is an effortto relate Smarandachely super119898minus119898119890119886119899 labeling and its dualfor 119898 ge 2 All results reported here are in total graph of pathand cycle119879(119875

119899) and119879(119862

119899) In future it is not only possible to

investigate some more results corresponding to other graphfamilies but also Smarandachely super m-mean labeling ingeneral as well

Disclosure

An earlier version of this paper was presented as an abstractat Distace in Graph 2016

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to express their very great apprecia-tion to Dr I Wayan Sudarsana Mrs Selvy Musdalifah andMrs Nurhasanah DaengMangesa for their valuable and con-structive suggestion during the conducting of this researchwork Their willingness to give their time so generously hasbeen very much appreciated

References

[1] F Harary ldquoRecent results on generalized Ramsey theory forgraphsrdquo in Graph Theory and Applications vol 303 of LectureNotes in Mathematics pp 125ndash138 Springer Berlin HeidelbergBerlin Heidelberg 1972

[2] S Somasundaram and R Ponraj ldquoMean labelings of graphsrdquoNational Academy of Science Letters vol 26 no 7-8 pp 210ndash213 2003

[3] R Ponraj and D Ramya ldquoOn super mean graphs of orderrdquoBulletin of Pure amp Applied Sciences vol 25 no 1 pp 143ndash1482006

[4] S Somasundaram and R Ponraj ldquoNon-existence of meanlabeling for a wheelrdquo Bulletin of Pure amp Applied Sciences-Mathematics and Statistics vol 22E pp 103ndash111 2003

[5] S Somasundaram and R Ponraj ldquoSome results on meangraphsrdquo Pure and Applied Mathematics Journal vol 58 pp 29ndash35 2003

[6] P Jeyanthi and D Ramya ldquoSuper mean labeling of some classesof graphsrdquo International Journal of Mathematical Combina-torics vol 1 pp 83ndash91 2012

[7] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meangraphsrdquo AKCE International Journal of Graphs and Combina-torics vol 6 no 1 pp 103ndash112 2009

[8] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meanlabeling of some graphsrdquo SUT Journal of Mathematics vol 46no 1 pp 53ndash66 2010

[9] P Jeyanthi and D Ramya ldquoSuper mean graphsrdquoUtilitas Mathe-matica vol 96 pp 101ndash109 2015

[10] D Ramya R Ponraj and P Jeyanthi ldquoSuper mean labeling ofgraphsrdquo Ars Combinatoria vol 112 pp 65ndash72 2013

[11] J A Gallian ldquoA dynamic survey of graph labellingsrdquo TheElectronic Journal of Combinatorics 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

International Journal of Mathematics and Mathematical Sciences 5

(iii) For even 119899 ge 6 if

1205821(V119894) =

6119899 119894 = 16119899 minus 3119894 + 4 119894 = 2 36119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 + 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 16119899 minus 6 119894 = 119899

1205821(119906119894) =

2119899 119894 = 12119899 minus 3119894 + 4 119894 = 2 32119899 minus 4119894 + 8 119894 = 4 5 6 1198992 + 12119899 minus 4119894 minus 7 119894 = 1198992 + 2 1198993 + 3 119899 minus 12119899 minus 6 119894 = 119899

(12)

then 1205821is a super mean labeling for 119879(119862

119899)

5 Summary and Remarks

Here we propose new results corresponding to super meanlabeling for total graph of path and cycleThiswork is an effortto relate Smarandachely super119898minus119898119890119886119899 labeling and its dualfor 119898 ge 2 All results reported here are in total graph of pathand cycle119879(119875

119899) and119879(119862

119899) In future it is not only possible to

investigate some more results corresponding to other graphfamilies but also Smarandachely super m-mean labeling ingeneral as well

Disclosure

An earlier version of this paper was presented as an abstractat Distace in Graph 2016

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to express their very great apprecia-tion to Dr I Wayan Sudarsana Mrs Selvy Musdalifah andMrs Nurhasanah DaengMangesa for their valuable and con-structive suggestion during the conducting of this researchwork Their willingness to give their time so generously hasbeen very much appreciated

References

[1] F Harary ldquoRecent results on generalized Ramsey theory forgraphsrdquo in Graph Theory and Applications vol 303 of LectureNotes in Mathematics pp 125ndash138 Springer Berlin HeidelbergBerlin Heidelberg 1972

[2] S Somasundaram and R Ponraj ldquoMean labelings of graphsrdquoNational Academy of Science Letters vol 26 no 7-8 pp 210ndash213 2003

[3] R Ponraj and D Ramya ldquoOn super mean graphs of orderrdquoBulletin of Pure amp Applied Sciences vol 25 no 1 pp 143ndash1482006

[4] S Somasundaram and R Ponraj ldquoNon-existence of meanlabeling for a wheelrdquo Bulletin of Pure amp Applied Sciences-Mathematics and Statistics vol 22E pp 103ndash111 2003

[5] S Somasundaram and R Ponraj ldquoSome results on meangraphsrdquo Pure and Applied Mathematics Journal vol 58 pp 29ndash35 2003

[6] P Jeyanthi and D Ramya ldquoSuper mean labeling of some classesof graphsrdquo International Journal of Mathematical Combina-torics vol 1 pp 83ndash91 2012

[7] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meangraphsrdquo AKCE International Journal of Graphs and Combina-torics vol 6 no 1 pp 103ndash112 2009

[8] P Jeyanthi D Ramya and P Thangavelu ldquoOn super meanlabeling of some graphsrdquo SUT Journal of Mathematics vol 46no 1 pp 53ndash66 2010

[9] P Jeyanthi and D Ramya ldquoSuper mean graphsrdquoUtilitas Mathe-matica vol 96 pp 101ndash109 2015

[10] D Ramya R Ponraj and P Jeyanthi ldquoSuper mean labeling ofgraphsrdquo Ars Combinatoria vol 112 pp 65ndash72 2013

[11] J A Gallian ldquoA dynamic survey of graph labellingsrdquo TheElectronic Journal of Combinatorics 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom