Cube difference labeling of some helm and wheel related graphs

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Cube difference labeling of some helm and

wheel related graphs

G. Amuda

1

, S. Meena

2

1. Lecturer in Mathematics Government College for Women (A),

Kumbakonam, Tamil Nadu, India.

2. Associate Professor and Head Department of Mathematics Government Arts College,.

C. Mutlur, Chidambaram, Tamil Nadu, India.

ABSTRACT

A Let

G

be a



p q

,



graph.

G

is said to admit a cube difference labeling if there exists an injection

  



:

0,1, 2,...,

1 f V G



p



such that the edges set of

G

has assigned a weight defined by the absolute cube

difference of its end-vertices, the resulting weights are distinct. A graph which admits cube difference labeling is called cube difference graph.In this chapter we investigate the existence of cube difference labeling of some helm and wheel related graphs, such as switching of rim vertices of wheel, apex vertex in helm, duplication of vertices of wheel and helm.

KEYWORDS - Cube difference labeling , Cube difference graph.

I. INTRODUCTION

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Definition: 1.1: Let 𝐺 = (𝑉, 𝐸) be a graph. A difference labeling of G is an injection f from V to the set of non- negative integer with weight function𝑓 ∗ on 𝐸 given by 𝑓 ∗ 𝑢𝑣 = │𝑓 𝑢 − 𝑓(𝑣)│for every𝑢𝑣edge in 𝐺.A graph with a difference labeling defined on it is called a labeled graph.

Definition: 1.2: Let

G





V E

,



be a graph .G is said to be cube difference labeling if there exist a bijection

f:V(G)→{0,1,2,….p-1} such that the induced function

f



:

E G

 



N

given by

 

3

 

3

f



uv





_

f u



_





_

f v



_

is injective.

Definition: 1.3: A vertex switching

G

_vof a graph

G

is the graph obtained by taking a vertex

v

of

G

, removing all

the edges to

v

and adding edges joining

v

to every other vertex which are not adjacent to

v

in

G

.

Definition: 1.4: A helm,

H n

_n

,



3

is the graph obtained from the wheel

W

_nby adding a pendant edge at each vertex on the wheel’s rim.

Definition 1.5: The wheel graph

W

_nis join of the graphs

C

_n and

K

₁. i.e.

W

_n

 

C

_n

K

₁. Here vertices

corresponding to

C

_nare called rim vertices and

C

_nis called rim of n W while the vertex corresponds to

K

₁is called

apex vertex.

Definition 1.6: Duplication of a vertex

v

_kof a graph G produces a new graph

G

₁by adding a vertex

v

_k' with

 

'

 

k k

N v



N v

. In other words a vertex

v

_k' is said to be a duplication of

v

_kif all the vertices which are adjacent

to

v

_kare now adjacent to

v

_k' also Duplication of a vertex

v

_kby a new edge

e



v v

_{k k}' "in a graph G produces a new

graph

G

' such that

N v

   

_k'



v v

_k

,

"_k and

Nv

"_k



 

v v

_k

,

_k' .

Definition 1.7: Duplication of an edge e = uvof a graph G produces a new graph G' by adding an edge e' = u' v ' such thatN(u ') = N(u)



(v ') -{v} and N(v') = N(v)



(u') -{u}.

Duplication of anedge e = uvby a new vertex w in a graph G produces a new graph G' such that N(w) = {u, v}.

II. MAIN RESULTS

Theorem: 2.1. For

n



3

, the graph obtained by switching any rim vertex of wheel

W

_n admits a cube difference

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Proof:

Let

u

be a apex vertex and let

u u

₁

,

₂

,...,

u

_n be rim vertices of

W

_n and 1

u

G

be the graph obtained by

switching a rim vertex

u

₁ of

W

_n.

Here

 

1

u

V G

 

n

and

 

1

3

5

u

E G

 

n

.

Define a vertex labeling

 





1

:

_u

0,1, 2,...,

f E G



n

as follows.

 

i

f u



i

for

1  

i

n

 

0 f u



We have to prove that the induced edge labeling function

 

1

:

_u

f



E G



N

defined by

 

3

 

3

f



uv



f u



f v

for all

uv



E G

 

is injective. The edge labelings are





₃





3

1

i i

f



u u

_

  

i

for

2   

i

n

1 

2



19,37,...,3

n

3 n

1 













3

1 i 1

1

1 f



u u

_

  

i

for

2   

i

n

2 





3



26, 63,...,

n

1

1 



 

3

i

f



uu



i

for

2  

i

n



_{8, 27,...,}

_n

3





Thus all the edge labels are distinct. Hence the graph obtained by switching any rim vertex of wheel graph

W

_n

admits a cube difference labeling.

Theorem: 2.2.The graph obtained by switching the apex vertex of helm

H

_n admits a cube difference labeling.

Proof:

Let

u

be a apex vertex and let

u u

₁

,

₂

,...,

u

_n be the rim vertices and

v v

₁

,

₂

,...,

v

_n be the pendant vertices of

helm

H

_n.

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Here

V G

 

_u



2 n



1

and

E G

 

_u



3 n

.

f

:

E G

  

_u



0,1,..., 2

n



as follows

 

i

f u



i

for

1  

i

n

 

i

f v

 

n

i

for

1  

i

n

 

0 f u



f

:

E G

 

u

N



_

defined by

 

3

 

3

f



uv



f u



f v

for all

uv



E G

 

is injective.

The edge labelings are





₃





3

1

i i

f



u u

_

  

i

for

1   

i

n

1 

2



7,19,...,3

n

3 n

1 









3

1

n

f



u u

 

n

 

3

i

f



uu



i

for

1  

i

n



3



1,8,...,

n



  



3

i

f



uv

 

n i

for

1  

i

n



 



 



3 3 3



1 ,

2 ,..., 2

n





 

₃





3

i i

f



u v

  

i

n i

for

1  

i

n











3 3 ₃ ₃



1 1,

2 2 ,..., 7

n











Thus all the edge labels are distinct. Hence the graph obtained by switching the apex vertex of helm

H

_n admits a

cube difference labeling.

Theorem: 2.3.The graph

G

obtained by duplicating all the vertices of the wheel graph.

W

_n, except the apex vertex

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Proof:

Let

G

be the graph obtained by duplicating all the vertices of wheel graph

W

_nexcept the apex vertex. Let

1 2

'

_,

'

_,...

'

n

u u

u

be the new vertices of

G

by duplicating

u u

₁

,

₂

,...,

u

_n. Then

V G

 





c u u

,

_i

,

_i'

1  

i

n



and

 



'











1 1 1

' '

,

1

1 ,

1

i i i i i _i i i

E G



cu cu

 

i

n



u u

_

   

i

n

u u

_

u u

_

  

i

n



1 ₁ 1



' '

,

n n n

u u u u u u



f V G

:

  



0,1, 2...2

n



as follows

 

_i

2

2 f u

 

i

for

1  

i

n

 

2 f c



n

 

'

2

1

i

f u

 

i

for

1  

i

n

f



:

E G

 



N

defined by

 

3

 

3

f



uv



f u



f v

for every

uv



E G

 

is injective.



 

  

3 3

1

2

i i

f



u u

_



i



i

for

  

i

n

1 

2



8,56,..., 24

n

72 n

56 







 



3

1

2

n

f



u u



n



    

3



3

2

i

f



cu



n



i



for

1  

i

n

   

  





3 3 ₃ 3 3



2 n

, 2

n

2 ,..., 2

n

2 n

2 



 

_'

  

3



3

2

1

i

f



cu



n



i



for

1  

i

n



₈

_n

3

_{1 ,8}

3

_n

3

_{3 ...,12}

3

_n

2

₆

_n

₁















 

3



3 1

'

2

1

i _i

f



u u

_



i



i



for

  

i

n

1 

_{27,117,...,36}

_n

2

₉₀

_n

₆₃



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 





3

1

'

2

1

n

f



u u



n









  

3 3

1

'

2

1

2

i i

f



u u

_



i



i

for

  

i

n

1 

2



7,37,...,12

n

30 n

19 





 





3

1

'

2

1

n

f



u u



n



Thus all the edge labels are distinct. Hence the graph obtained by duplicating all the vertices of the wheel graph

W

_n

except the apex vertex admits a cube difference labeling.

Theorem: 2.4. The graph obtained by duplicating all the vertices in the rim of helm

H

_n admits a cube difference

labeling. Proof:

Let

G



H

_n be the graph with vertex set

V G

 





c u v

, ,

_i _i

1  

i

n



and the edge set

 



i

,

i i

1  

i i 1

1

1 



,

n



.

E G



cu u v

 

i

n



u u

_

   

i

n

u u

Let

G

'be the graph obtained by duplicating all the rim vertices in

H

_n and let the new vertices be

u u

₁'

,

₂'

...

u

_n'

, then the vertex set

V G

  

'



c u v u

, , ,

_i _i _i'

1  

i

n



and the edge set

  





1 1 1



' ' ' ' '

,

1 ,

_i

,

1

i

i i i i i i i i i i

E G

cu cu u v v u

i

n

u u

_

u u

_

i

n





 



  



1 1



' ' 1

,

n n n

u u u u u u



Here

V G

 

'

 

3 n

1

and

E G

 

'



7 n

.

f V G

:

 

'





0,1, 2...3

n



as follows

 

_i

1 f u

 

i

for

1  

i

n

 

i

2

1 f v

  

n

i

for

1  

i

n

 

1

2

i

f u

 

n

i

for

1  

i

n

 

f c



n

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 

3

 

3

f



uv



f u



f v

for every

uv



E G

 

is injective.

 

₃





3

1

i

f



cu

  

n

i

for

1  

i

n



3 3 2



,

1,...,3

3

1 n n

n







  

 

3



3

1

2

1

i i

f



u v

 

i

  

n

i

for

1  

i

n



 





3 3 3 3



1 ,

3 1,..., 26

24

6 n

n











 



3 ₃

1

i i

f



u u

_

 

i



i

for

1   

i

n

1 

2



1, 7,...,3

n

9 n

7 













3

1

n

f



u u

 

n

 

_' ₃





3

2

i

f



cu

 

n



i

for

1  

i

n











3 ₃ 3 ₃ ₃



2 ,

4 ,..., 26

n











 

'



 

3



3

2

1

2

i i

f



v u

  

n

i

 

n

i

for

1  

i

n



 



  





3 3 3 3 3 3



2 1 ,

4 3 ,..., 3

3

1 n

n





 



 





'



  

3



3

1

2

i i

f



u u

_

 

i

  

n

i

for

1   

i

n

1 

 



  





3 3 3 3



4 ,

6 1,..., 3

2 n

n







 

 

1

' 2

9

n

f



u u



n



n





_'







3 ₃

1

2

i i

f



u u

_

 

n

i



i

for

1   

i

n

1 













3 3 3 3 3



2 1,

4 2 ,..., 3

2 n

n











 

_'

 

3

1

3

n

f



u u



n

(8)

779 |

P a g e

n

H

Theorem: 2.5.The graph

G

obtained by duplicating all the vertices of the helm

H

_n, except the apex vertex admits

a cube difference labeling. Proof:

Let

G



H

_nbe the graph with vertex set

V G

 





c u v

, ,

_i _i

1  

i

n



and the edge set

 



i

,

i i

1  

i i 1

1

1 

 

1 n

E G



cu u v

 

i

n



u u

_

   

i

n

u u

.

Let

G

' be the graph obtained by duplicating all the vertices in the helm

H

_n, except the apex vertex

c

.

Let

u u

₁'

,

₂'

...

u

_n' and

v v

₁'

, ...

₂'

v

_n' be the new vertices of

G

by duplicating

u u

₁

,

₂

...

u

_n and

v v

₁

, ...

₂

v

_n then

the vertex set

V G

  

'



c u v u v

, , , ,

_i _i _i' _i'

1  

i

n



and the edge set

  





1 ₁ 1



' ' ' ' '

,

1 ,

,

1

i i i i i i i i i i i _i i i

E G

cu cu u v v u u v

i

n

u u

_

u u

i

n

 



 



  



1 1



' ' 1

,

n n n

u u u u u u



Here

V G

 

'

 

4 n

1

and

E G

 

'



8 n

.

f V G

:

 

'





0,1, 2...4

n



as follows

 

f c



n

 

_i

1 f u

 

i

for

1  

i

n

 

i

f v

 

n i

for

1  

i

n

 

'

2

i

f u

  

n

i

for

1  

i

n

 

'

2

3

i

f v

  

n

i

for

1  

i

n

f



:

E G

 

'



N

defined by

 

3

 

3

f



uv



f u



f v

for every

uv



E G

 

is injective.

 

3





3

1

i

(9)

780 |

P a g e



3 3 2



,

1,...,3

3

1 n n

n







 



 

3



3

1

i i

f



u v

 

i

 

n i

for

1  

i

n



 



  





3 3 3 3



1 ,

2 1,..., 2

1 n

n







 



 



3 ₃

1

i i

f



u u

_

 

i



i

for

1   

i

n

1 

2



1, 7,...,3

n

9 n

7 













3

1

n

f



u u

 

n

 

_' ₃





3

2

i

f



cu

   

n

i

for

1  

i

n















3 3 3 3 3 3



4 ,

6 ,..., 3

2 n

n















 

_'



 

3



3

1

2

3

i i

f



u v

 

i

  

n

i

for

1  

i

n



 





 





3 3 ₃ 3 3



5 ,

7 1 ,..., 3

3

1 n

n









 



_'



  

3



3

1

2

4

i i

f



u u

_

 

i

  

n

i

for

1   

i

n

1 

 





 





3 3 3 3 3



6 ,

8 1 ,..., 3

2

2 n

n









 

 

₁_'



 

3



3

1

4

n

f



u u

 

n



n





_'



₃





3

1

2

i i

f



u u

_

   

i

n

i

for

1   

i

n

1 







  





3 ₃ 3 ₃ 3 3



4 1 ,

6 2 ,..., 3

1 n

n











 

 

_'





3

1 n

3

2 f



u u



n



 

_'



 

3



3

2

i i

f



v u

 

n i



n

 

i

for

1  

i

n



 





  



3 3 3 3 3 3



4 1 ,

6 2 ,..., 3

2

2 n

n





 



 





(10)

781 |

P a g e

n

H

III. Conclusion

It is very exciting to investigate on graph families which are cube difference graphs. Those results are seems easy but it is a big challenge and good experience for us to investigate and prove. Here we have analyzed and proved in details about some results on cube difference graphs such as helm and wheel related graphs such as switching of wheel and helm, duplication of helm and except apex vertex of helm graph are cube difference.

IV. ACKNOWLEDGEMENT

The author is grateful to my guide and thanks to ManonmaniamSundaranar University for providing facilities. I wish

to express my deep sense of gratitude to Dr. M. Karuppaiyan my husband for their immense help in studies.

REFERENCES

[1] V. Ajitha, S. Arumugam, and K. A. Germina. “On Square sum graphs” AKCE J. Graphs, Combin, 6, 2006. 1- 10.

[2] Frank Harary. “Graph theory” Narosa Publishing House 2001.

[3] J. A. Gallian. “A dynamic survey of graph labeling” The Electronics journal of Combinatories, 17, 2010 # DS6. [4] J. Shiama. “Square sum labeling for some middle and total graphs” International Journal of Computer

Applications (0975-08887), 37(4), January 2012.

[5] J. Shiama. “Square difference labeling for some graphs” International Journal of Computer Applications (0975-08887), 44 (4), April 2012.

[6] J. Shiama. “Some Special types of Square difference graphs” International Journal of Mathematical archives, 3(6), 2012, 2369-2374 ISSN 2229-5046.

[7] J. Shiama. “Cube Difference Labeling Of Some Graphs” International Journal of Engineering Science and Innovative Technology, 2(6), November 2013.