Subdivision of Harary graph: A scientific approach …bosaljournals.com/csp/images/pdffiles/48CSP.pdfarithmetic progression with initial term a and difference d. The total labeling

ISSN: 2410-8790 Bhatti et al / Current Science Perspectives 3(4) (2017) 156-164 iscientic.org.

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Article type: Research article Article history: Received July 2017 Accepted August 2017 October 2017 Issue Keywords: Harary graph

Super ),( da -EAT

Landscape connectivity

Subdivision of Harary graph

Graph order p

Graph size

Graph structures have been exposed to be a dominant and helpful way of modeling landscape networks. Labeling of harary graphs is an easy scientific approach towards landscape connectivity. In this paper, the super (a,d)-edge-antimagic total labeling of subdivided Harary graphs ht

pC , is investigated for the

even 2h . We segregated the subdivided Harary graph into two cases. In first

case, when the order of the subdivided Harary graphs p varies then the distance t will remain same while in other case, when the order p varies then the distance t will also vary.

© 2017 International Scientific Organization: All rights reserved.

Capsule Summary: This work focuses on mapping of landscape connectivity by making use of subdivision of harary graph through super edge antimagic total labeling.

Cite This Article As: Akhlaq Ahmad Bhatti, Khalid Arif, Faisal Yasin and Iftikhar Ahmad. Subdivision of Harary graph: A scientific approach towards landscape connectivity. Current Science Perspectives 3(4) (2017) 156-164

INTRODUCTION Numerous scientific approaches enable measuring and mapping connectivity for a given population or species. However, the most important approach is to find out the effective passage for refurbishing connectivity. Landscapes or networks connect the people in many ways. The arrangement of a network along with its nodes and connecting lines is worth noticing as it is one of the growing property which affects humanity in various ways. A graph represents landscape as a set of nodes and edges. Graph theory uses quick algorithms and compact data construction that are easily modified to landscape connectivity for studying main species in (Fall et al., 2007; Minor & Urban 2007; Saura & Pascual-Hortal, 2007). Landscapes can be examined as a

system of environment patches related with scattering individuals. All graphs in this paper are finite, simple and undirected. The graph G has the vertex-set )(GV and edge-set )(GE . A graph

labeling is a mapping that assigns numbers to graph elements. In this paper the domain will be the set of all vertices, set of all edges or set of all vertices and edges. Labeling in which domain is set of vertices and edges is called total labeling. A total labeling is called ),( da -edge antimagic

total labeling if the sum of edge weights constitutes an arithmetic progression with initial term a and difference d .

The total labeling is called super ),( da -edge antimagic total if

the smallest labels are assigned to vertices. There are many types of graph labelings, for example harmonius, cor-dial,

Current Science Perspectives 3(4) (2017) 156-164

Subdivision of Harary graph: A scientific approach towards landscape connectivity

Akhlaq Ahmad Bhatti1, Khalid Arif *1, 2, Faisal Yasin2 and Iftikhar Ahmad3

1Department of Sciences & Humanities, National University of Computer and Emerging Sciences, Lahore, Pakistan 2Department of Mathematics & Statistics, The University of Lahore, Lahore-Pakistan

3Department of Mathematical Sciences, Göteborg University, SE-41296, Göteborg, Sweden *Corresponding Author’s Email: [email protected]

A R T I C L E I N F O A B S T R A C T

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graceful and antimagic. In this paper, we focus on one type of labelings called super ),( da -edge antimagic total labeling.

Super magic labeling was introduced by Stewart (1966). For

),( EV graph G , a bijective mapping )()(: GEGVf

|}|||,{1,2,3, EV is an edge-magic total labeling of G if

)(=)()()( constantkyfxyfxf , where k is a constant,

independent of the choice of edge )(GExy . The subject of

edge-magic total labeling of graphs has its origin in the work of Kotzig & Rosa (1970) on what they called magic valuations (Sedlacek, 1963; Kotzig & Rosa, 1972; Sedlacek, 1976). But most of the work on it is found in Ringel & Lládo (1996). The concept of Super Edge-Magic Total labeling of a graph G is defined in Enomoto et al. (1998) and Ali et al. (2016) as a bijective function |}|||,{1,2,3,)()(: EVGEGVf , such that

in addition to being an edge-magic total labeling of G , if it satisfies the extra property that is },{1,2,3,=))(( vGVf .

Wallis calls this labeling strongly edge-magic. The concept of an antimagic labeling was introduced (Hartsfield & Ringel, 1989; Hartsfield & Ringel, 1990). In their terminology, a graph G is called antimagic if its edges are labeled with labels },{1,2,3, e in such a way that all vertex-

weights are pairwise distinct, where a vertex-weight of a vertex v is the sum of labels of the entire edges incident withv . An ),( da -EAT labeling of a graph G is defined as a one-to-one

mapping f from )()( GEGV to the set },{1,2,3, ev so that

the set of edge-weights )}(:)()()({ GExyyfxyfxf equals

}1)(,,2,,{ deadadaa , for two integers 0>a and

0d . Notice that the same labeling would be an edge-magic total labeling when 0=d . In other words ,0)(a -EAT labeling is

an EMT labeling of G .

An ),( da -EAT labeling is called super if the smallest labels

appear on the vertices of G , i.e. },{1,2,3,=))(( vGVf (Bhatti &

Javaid, 2013). The definition of ),( da -EAT labeling was

introduced (Simanjantuk & Miller, 2000; Simanjantuk et al., 2000). The ),( da -edge antimagic total labeling and super ),( da

-edge antimagic total labelings are natural extensions of the notion of an edge-magic total labeling, defined by Kotzig & Rosa (1970) and notion of super edge-magic total labeling which was defined by Enomoto et al. (1998).

Super edge-antimagic total labeling for Harary graphs t

pC

was constructed by Hussain et al. (2012). They worked on super ),( da -edge antimagic total labeling and super (a, d)-

vertex antimagic total labeling. They also constructed the super edge-antimagic and super vertex-antimagic total

labelings for a disjoint union of k identical copies of the

Harary graph. Baskoro et al. (2007) show how to construct new larger super ),( da -edge antimagic total graphs from

existing smaller one. The concept of antimagic labeling of the union of subdivided stars and antimagic labeling of antiprisms were introduced (Raheem & Baig, 2016; Baca & Martin, 2011). Super (a, d) edge antimagic total labeling and super (a, d) edge magic total labeling of subdivided stars and

w-trees explained in (javaid et al., 2010; Salman et al., 2010; javaid et al., 2012; javaid et al., 2014; Bhatti et al., 2015; Delman & Koilraj, 2015). All trees with at most 17 vertices are super edgemagic (Lee & Shan, 2002). The labeling on subdivision of grid graphs using magic and antimagic labeling in (Tabraiz & Hussain, 2016).

MATERIAL AND METHODS

Harary Graph

For 2t and 4p , a Harary graph t

pC is a graph constructed

from a cycle pC by joining any two vertices at distance t in

pC .

Subdivision of Harary Graph

For 6t and 16p , a subdivided Harary graph ht

pC , is a

graph constructed from Harary graph t

pC after the

subdivision (for even 2h ) of each edge of grapht

pC .

RESULTS AND DISCUSSION

We prove that subdivided Harary graph is a super ),( da -edge

antimagic total labeling for even 2h .

Theorem 1 For any 25p with 2=h and for 6=t , 6,2

pCG

admits a super 2,1)(2 p edge-antimagic total labeling.

Proof Let us denote the vertex and edge set of G as pGV |=)(| and qGE |=)(| . Let )(),( 11 GEGV and )(),( 22 GEGV denote the

vertices on the outer and inner cycle respectively. The vertex [ )()(=)( 21 GVGVGV ] and edge [ )()(=)( 21 GEGEGE ] sets

of G are defined as follows:

},,15

1:{}5

31:{=)( hj

piv

pivGV j

i

}5

1:{}5

31:{=)( 1

231

pivv

pivvGE iii

1},15

1:{ 1 hjp

ivv jj

},5

1:{p

ivvh

Where titii 23=,22,33= and all indices are

taken in mod 5

3 p . Now we define labeling

},{1,2,3,...)()(: qpGEGV . We label the vertices on

outer and inner cycle as follows:

=)( iv

1.5

31,1

5

;5

3=1,

5

2

pifori

p

pifor

pi





=)( jv

2.=,5

15

,15

2

2;=2,5

1,15

1;=1,2=,5

1,)(2

jp

ip

forip

jp

iforip

kkjp

iforikp

We label the edges on outer and inner cycle as follows:

=)( 1iivv

2.5

31,1

;5

31

5

3,1

5

8

piforiqp

pi

pforiq

p

.5

,11=)( 1

23

piiqvv i

1.=1,2=,5

,1)(1=)( 1 kkjp

iikqvv jj

=)( vvh

.5

15

,153

4

2;5

1,13

4

pi

pfori

pq

pifori

q

Edge weights of all edges in )(1 GE will form consecutive

integers 15

133,...,2,22

ppp , where the weight 22 p is

obtained by the edge 15,1

5

3 vv h

p

if tp

8

3 . Edge weights of all

edges in )(2 GE will form consecutive integers

15

163,...,

5

132,

5

13

ppp . Therefore, all the edge weights form

consecutive integers 15

163,...,2,22

ppp . Since all vertices

receive smallest labels so is a super 2,1)(2 p edge

antimagic total labeling.

Figure 1 shows the super 2,1)(2 p -EAT total labeling of 6,2

40C

. Theorem 2 For any 45p with 4=h and for 10=t ,

10,4

pCG admits a super 2,1)(2 p edge-antimagic total

labeling. Proof Let us denote the vertex and edge set of G as pGV |=)(|

and qGE |=)(| . Let )(),( 11 GEGV and )(),( 22 GEGV denote the

vertices on the outer and inner cycle respectively. The vertex

)]()(=)([ 21 GVGVGV and edge [ )()(=)( 21 GEGEGE ]

sets of G are defined as follows:

},,19

1:{}9

51:{=)( hj

piv

pivGV j

i

}9

1:{}9

51:{=)( 1

451

pivv

pivvGE iii

1},19

1:{ 1 hjp

ivv jj

},9

1:{p

ivvh

where titii 45=,44,55= and all indices are

taken in mod 9




=)( iv

2.9

51,2

9

2

;9

51

9

52,

3p

iip

pi

ppi

=)( jv

4.=,9

19

,239

4

4;=2,9

1,239

2

1;=,2=,9

1,2)(39

2

1,2;=1,2=,9

1,2)(3

jp

ip

ip

jp

iip

kkjp

iikp

kkjp

iikp


=)( 1iivv

3.9

51,2

;9

52

9

5,2

9

14

piforiqp

pi

piq

p

.9

,12110

11=)( 1

45

pii

qvv i

=)( 1jj vv

1.=,2=,9

1,2)(110

9

1,2;=1,2=,9

1,2)(210

11

kkjp

iikq

kkjp

iikq

=)( vvh

.9

19

,195

7

2;9

1,15

7

pi

pi

pq

pii

q






integers 19

233,...,2,22


obtained by the edge 1

9,19

5 vv h

p

if tp

14

5 . Edge weights of all

edges in )(2 GE will form consecutive integers

19

283,...,

9

232,

9

23

ppp . Therefore, all the edge weights

form consecutive integers 19

283,...,2,22

ppp . Since all

vertices receive smallest labels so is a super 2,1)(2 p

edge antimagic total labeling. Figure 2 shows the super 2,1)(2 p -EAT total labeling of

10,4

45C.

Theorem 3 For any 520 np , with 1,2= nnh and for

12,4= nnt , nn

pCG 2,24 admits a super 2,1)(2 p edge-

antimagic total labeling. Proof Let us denote the vertex and edge set of G as

pGV |=)(| and qGE |=)(| . Let )(),( 11 GEGV and )(),( 22 GEGV

denote the vertices on the outer and inner cycle respectively.

Figure 1: super - EAT total labeling of






The vertex [ )()(=)( 21 GVGVGV ] and edge [

)()(=)( 21 GEGEGE ] sets of G are defined as follows:

1},14

1)(21:{=)(

n

n

pnivGV i

1},,,114

1:{

nhjn

piv j

1},14

1)(21:{=)( 1

n

n

pnivvGE ii

1},14

1:{ 1

21)(2

nn

pivv nin

1}1,,114

1:{ 1

nhjn

pivv jj

1},,14

1:{

nn

pivvh

Where

tnintninnin 21)(2=,21)(2,21)(2= and all indices are taken in mod 1,

14

1)(2

n

n

pn . Now we

define labeling },{1,2,3,...)()(: qpGEGV . We label

the vertices on outer and inner cycle as follows:

=)( iv

1.,14

1)(21,

14

1;,14

1)(21)(

14

1)(2,

14

1)(

nnn

pniin

n

np

nn

pnin

n

pnn

n

pni

=)( jv

1,,=,14

114

,14

2

1;,=2,14

1,14

2;1,,12=,14

1,14

1;,1,12=,14

1,

nhjn

pi

n

p

n

np

nhjn

pi

n

np

nnkkjn

pi

n

np

nnkkjn

pip

Where nikn 1)(= and nin 1)(2= . We label

the edges on outer and inner cycle as follows:

=)( 1iivv

1.1),(14

1)(21,

1;,14

1)(2

14

1)(2,

14

2)(6

nnn

pniinqp

nn

pnin

n

pninq

n

pn

1.,14

,1124

1)(5=)( 1

21)(2

n

n

pini

n

qnvv nin

=)( 1jj vv

2,1,,12=,1,24

1)(4

1;,1,12=,1,24

1)(5

nnkkjin

qn

nnkkjin

qn

Where niknnikn )1(=,)(= and .

14=

n

p

=)( vvh

1.,14

114

,11412

1)(3

1;2,14

1,112

1)(3

nn

pi

n

pi

n

p

n

qn

nn

pii

n

qn


integers 11,14

3)(103,...,2,22

n

n

pnpp , where the weight

22 p is obtained by the edge ,11),1(4

14

1)(2 vvh

nn

pn

if

1,26

1)(2

nt

n

pn . Edge weights of all edges in )(2 GE will

form consecutive integers

11,14

4)(123,...,

14

3)(102,

14

3)(10

n

n

pn

n

pn

n

pn .

Therefore, all the edge weights form consecutive integers

11,14

4)(123,...,2,22

n

n

pnpp . Since all vertices


antimagic total labeling.

Theorem 4 For any even p , 16p with 2=h and for any

t (which is multiple of 3 ), 6t , ,2t

pCG admits a super

2,1)(2 p edge-antimagic total labeling.

Proof Let us denote the vertex and edge set of G as pGV |=)(| and qGE |=)(| . Let )(),( 11 GEGV and )(),( 22 GEGV

denote the vertices on the outer and inner cycle respectively. The vertex [ )()(=)( 21 GVGVGV ] and edge [


},,18

1:{}4

31:{=)( hj

piv

pivGV j

i

}8

1:{}4

31:{=)( 1

231

pivv

pivvGE iii

1},18

1:{ 1 hjp

ivv jj

},8

1:{p

ivvh


taken in mod 4








=)( iv

1.4

31,1

8

;4

3=1,

8

5

pifori

p

pifor

pi

=)( jv

1.=,2=,8

1,)(28

1;=1,2=,8

1,)(2

kkjp

iforikp

kkjp

iforikp


=)( 1iivv

2.4

31,1

;4

31

4

3,1

4

7

piforiqp

pi

pforiq

p

.8

,11=)( 1

23

piiqvv i

1.=1,2=,8

,1)(1=)( 1 kkjp

iikqvv jj

.8

,119

11=)(

pii

qvvh


integers 18

193,...,2,22


obtained by the edge 1

8

31

8

31,

p

h

p vv if tp

=8

3 . Edge weights of

all edges in )(2 GE will form consecutive integers

18

253,...,

8

192,

8

19



253,...,2,22

ppp . Since all



12,2

32C.

Theorem 5 For any even p , 28p with 4=h and for any

t (which is multiple of 5 ), 10t , ,4t

pCG admits a super

2,1)(2 p edge-antimagic total labeling.

Proof Let us denote the vertex and edge set of G as pGV |=)(| and qGE |=)(| . Let )(),( 11 GEGV and )(),( 22 GEGV denote

the vertices on the outer and inner cycle respectively. The vertex [ )()(=)( 21 GVGVGV ] and edge [


},,114

1:{}7

51:{=)( hj

piv

pivGV j

i

}14

1:{}7

51:{=)( 1

451

pivv

pivvGE iii

1},114

1:{ 1 hjp

ivv jj

},14

1:{p

ivvh


taken in mod 7




=)( iv

2.7

51,2

7

;7

51

7

52,

7

4

pifori

p

pi

pfor

pi

=)( jv

1,2.=,2=,14

1,2)(37

1,2;=1,2=,14

1,2)(3

kkjp

iforikp

kkjp

iforikp


=)( 1iivv

3.7

51,2

;7

52

7

5,2

7

12

piforiqp

pi

pforiq

p

.14

,12115

16=)( 1

45

pii

qvv i

=)( 1jj vv

1.=,2=,14

1,2)(115

14

1,2;=1,2=,14

1,2)(215

16

kkjp

iforikq

kkjp

iforikq

.14

,1115

19=)(

pii

qvvh


integers 114

333,...,2,22

ppp , where the weight 22 p

is obtained by the edge 1

14

51

14

51,

p

h

p vv if tp

=14

5 . Edge weights

of all edges in )(2 GE will form consecutive integers

114

433,...,

14

332,

14

33







433,...,2,22

ppp . Since all



15,4

42C.

Theorem 6 For any even p , 412 np with 1,2= nnh

and for any t (which is multiple of 12 n ), 12,4 nnt , nt

pCG ,2 admits a super 2,1)(2 p edge-antimagic total

labeling.

Proof Let us denote the vertex and edge set of G as pGV |=)(| and qGE |=)(| . Let )(),( 11 GEGV and

)(),( 22 GEGV denote the vertices on the outer and inner

cycle respectively. The vertex [ )()(=)( 21 GVGVGV ] and

edge [ )()(=)( 21 GEGEGE ] sets of G are defined as

follows:

1},13

1)(21:{=)(

n

n

pnivGV i

1},,,126

1:{

nhjn

piv j







1},13

1)(21:{=)( 1

n

n

pnivvGE ii

1},26

1:{ 1

21)(2

nn

pivv nin

1}1,,126

1:{ 1

nhjn

pivv jj

1},,26

1:{

nn

pivvh

Where

tnintninnin 21)(2=,21)(2,21)(2= and all indices are taken in mod

1,13

1)(2

n

n

pn . Now we

define labeling },{1,2,3,...)()(: qpGEGV . We label

the vertices on outer and inner cycle as follows:

=)( iv

1.,13

1)(21,

26

1;,13

1)(21)(

13

1)(2,

26

2)(3

nnn

pniin

n

np

nn

pnin

n

pnn

n

pni

=)( jv

1,,1,2=,26

1,26

1;,11,2=,26

1,

nnkkjn

pi

n

np

nnkkjn

pip

Where nikn 1)(= . We label the edges on outer and

inner cycle as follows:

=)( 1iivv

1.1),(13

1)(21,

1;,13

1)(2

13

1)(2,

13

2)(5

nnn

pniinqp

nn

pnin

n

pninq

n

pn

1.,26

,1136

2)(7=)( 1

21)(2

n

n

pini

n

qnvv nin

=)( 1jj vv

2,1,1,2=,26

1,36

2)(6

1;,11,2=,26

1,36

2)(7

nnkkjn

pi

n

qn

nnkkjn

pi

n

qn

Where nikn )(= and .1)(= nikn

1,26

,1136

3)(8=)(

n

n

pii

n

qnvvh

.


integers 1,1,26

5)(143,...,2,22

n

n

pnpp where the

weight 22 p is obtained by the edge 1

26

1)(21

26

1)(21,

n

pn

h

n

pn vv if

1,=26

1)(2

nt

n

pn . Edge weights of all edges in )(2 GE will

form consecutive integers

11,26

7)(183,...,

26

5)(142,

26

5)(14

n

n

pn

n

pn

n

pn .

Therefore, all the edge weights form consecutive integers

11,26

7)(183,...,2,22

n

n

pnpp . Since all vertices


antimagic total labeling. CONCLUSIONS The graph theory is a reliable scientific approach for protection of environmental networks in much realistic way. The implementation of graph theory through subdivision of harary graph by antimagic total labeling facilitates to draw connectivity responses more authentically and to reduce uncertainty marks associated with previous models. It is concluded that harary graph has come out as one of the most imperative mathematical device for demonstration and analysis of the processes which are essentially sequential in nature. REFERENCES Ali, A., Javaid, M., Rehman, M.A. 2016, SEMT Labeling on

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