Degree Splitting of Root Square Mean Graphs

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http://dx.doi.org/10.4236/am.2015.66086

Degree Splitting of Root Square Mean

Graphs

S. S. Sandhya

1

, S. Somasundaram

2

, S. Anusa

3

1_{Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai, India} 2_{Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, India}

3_{Department of Mathematics, Arunachala College of Engineering for Women, Vellichanthai, India}

Email: anu12343s@gmail.com

Received 15 April 2015; accepted 30 May 2015; published 2 June 2015

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

Let f V G:

( ) {

→ 1, 2,,q+1

}

be an injective function. For a vertex labeling f, the induced edge

labeling f∗

(

e=uv

)

is defined by,

(

)

( )

 

 

2 2

2

f u f v

f∗ e=uv = + or

( )

;

 

 

2 2

2

f u + f v

then,

the edge labels are distinct and are from

{

1, 2,, q

}

. Then f is called a root square mean labeling of

G. In this paper, we prove root square mean labeling of some degree splitting graphs.

Keywords

Graph, Path, Cycle, Degree Splitting Graphs, Root Square Mean Graphs, Union of Graphs

1. Introduction

The graphs considered here are simple, finite and undirected. Let V G

( )

denote the vertex set and E G

( )

de-note the edge set of G. For detailed survey of graph labeling we refer to Gallian [1]. For all other standard ter-minology and notations we follow Harary [2]. The concept of mean labeling on degree splitting graph was in-troduced in [3]. Motivated by the authors we study the root square mean labeling on degree splitting graphs. Root square mean labeling was introduced in [4] and the root square mean labeling of some standard graphs was proved in [5]-[11]. The definitions and theorems are useful for our present study.

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each edge e=uv is labeled with

(

)

( )

2 2

2 f u f v f e uv

 ₊ 

 

= =

 

or

( )

2 2

2 f u f v

 ₊ 

 

 

, then the edge

labels are distinct and are from

{

1, 2,,q

}

. In this case f is called root square mean labeling of G.

Definition 1.2: A walk in which u u_{1 2}u_n are distinct is called a path. A path on n vertices is denoted by

n P .

Definition 1.3: A closed path is called a cycle. A cycle on n vertices is denoted by C_n.

Definition 1.4: Let G=

(

V E,

)

be a graph with V =S₁   S₂  S_t T, where each S_i is a set of vertices having at least two vertices and having the same degree and T= −V S_i. The degree splitting graph of G is denoted by DS G

( )

and is obtained from G by adding the vertices w w₁, ₂,,wt and joining wi to each vertex of Si,1≤ ≤i t. The graph G and its degree splitting graph DS G

( )

are given inFigure 1.

Definition 1.5: The union of two graphs G₁=

(

V E₁, ₁

)

and G₂ =

(

V E₂, ₂

)

is a graph G=G₁G₂ with vertex set V =V₁V₂ and the edge set E=E₁E₂.

Theorem 1.6: Any path is a root square mean graph. Theorem 1.7: Any cycle is a root square mean graph.

2. Main Results

Theorem 2.1: nDS P

( )

₃ is a root square mean graph. Proof: The graph DS P

( )

₃ is shown inFigure 2.

[image:2.595.80.540.81.128.2]

Let G=nDS P

( )

₃ . Let the vertex set of G be V =V₁  V₂  V_n where Vi=

{

v v v w₁i, ₂i, ₃i, i,1≤ ≤i n

}

. De-fine a function f V G:

( ) {

→ 1, 2,,q+1

}

by

Figure 1. The graph G and its degree splitting graph DS G

( )

.

[image:2.595.103.528.363.733.2]

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( )

1 4 3, 1 i

f v = −i ≤ ≤i n

( )

2 4 2, 1 i

f v = −i ≤ ≤i n

( )

3 4 1, 1 i

f v = −i ≤ ≤i n

( )

_i 4 , 1

f w = i ≤ ≤i n

Then the edges are labeled as

( )

1 2 4 3, 1 1

i i

f v v = −i ≤ ≤ −i n

( )

2 3 4 1, 1 1

i i

f v v = −i ≤ ≤ −i n

( )

1 4 2, 1 2

i i

f v w = −i ≤ ≤ −i n

( )

3 4 , 1 2

i i

f v w = i ≤ ≤ −i n

Then the edge labels are distinct and are from

{

1, 2,,q

}

. Hence by definition 1.1, G is a root square mean graph.

Example 2.2: Root square mean labeling of 4DS P

( )

₃ is shown inFigure 3. Theorem 2.3:4DS P

( )

₄ is a root square mean graph.

Proof: The graph DS P

( )

₄ is shown inFigure 4.

[image:3.595.153.479.378.733.2]

Let G=nDS P

( )

₃ . Let the vertex set of G be V =V₁  V₂  V_n where Vi=

{

v v v v w w₁i, ₂i, i₃, ₄i, ₁i, ₂i,1≤ ≤i n

}

. Define a function f V G:

( ) {

→ 1, 2,,q+1

}

by

Figure 3. Root square mean labeling of 4DS P

( )

₃ .

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( )

1 7 5, 1 i

f v = −i ≤ ≤i n

( )

2 7 3, 1 i

f v = −i ≤ ≤i n

( )

3 7 1, 1 i

f v = −i ≤ ≤i n

( )

4 7 4, 1 i

f v = −i ≤ ≤i n

( )

1 7 6, 1

i

f w = −i ≤ ≤i n

( )

2 7 , 1 i

f w = i ≤ ≤i n

( )

1 2 7 4, 1 i i

f v v = −i ≤ ≤i n

( )

2 3 7 2, 1 i i

f v v = −i ≤ ≤i n

( )

3 4 7 3, 1 i i

f v v = −i ≤ ≤i n

( )

1 1 7 6, 1 i i

f v w = −i ≤ ≤i n

( )

1 4 7 5, 1

i i

f w v = −i ≤ ≤i n

( )

2 2 7 1, 1 i i

f v w = −i ≤ ≤i n

( )

3 2 7 , 1 i i

f v w = i ≤ ≤i n

{

1, 2,,q

}

Example 2.4: Root square mean labeling of 4DS P

( )

₃ is shown inFigure 5. Theorem 2.5: nDS P

(

₂K₁

)

is a root square mean graph.

(

₂K₁

)

is shown inFigure 6.

Let G=nDS P

(

₂K₁

)

. Let the vertex set of G be V =V₁  V₂  V_n where

{

1, 2, 3, 4, 1, 2,1

}

i i i i i i i

V = v v v v w w ≤ ≤i n . Define a function f V G:

( ) {

→ 1, 2,,q+1

}

by

[image:4.595.135.477.472.701.2]

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[image:5.595.93.529.74.570.2]

Figure 6.The graph DS P

(

2K1

)

.

( )

1 7 5, 1 i

f v = −i ≤ ≤i n

( )

2 7 4, 1 i

f v = −i ≤ ≤i n

( )

3 7 2, 1 i

f v = −i ≤ ≤i n

( )

4 7 1, 1 i

f v = −i ≤ ≤i n

( )

1 7 6, 1

i

f w = −i ≤ ≤i n

( )

2 7 , 1 i

f w = i ≤ ≤i n

( )

1 3 7 4, 1 i i

f v v = −i ≤ ≤i n

( )

3 4 7 2, 1 i i

f v v = −i ≤ ≤i n

( )

4 2 7 3, 1 i i

f v v = −i ≤ ≤i n

( )

1 1 7 6, 1 i i

f v w = −i ≤ ≤i n

( )

2 1 7 5, 1

i i

f v w = −i ≤ ≤i n

( )

3 2 7 1, 1 i i

f v w = −i ≤ ≤i n

( )

4 2 7 , 1 i i

f v w = i ≤ ≤i n

{

1, 2,,q

}

Example 2.6: The labeling pattern of 4DS P

(

₂K₁

)

is shown inFigure 7. Theorem 2.7:nDS P

(

₂K₂

)

(

₂K₂

)

Let G=nDS P

(

₂K₂

)

{

1, 2, 3, 4, 5, 6, 1, 2,1

}

i i i i i i i i i

V = v v v v v v w w ≤ ≤i n . Define a function f V G:

( ) {

→ 1, 2,,q+1

}

by

( )

1 11 5, 1 i

f v = i− ≤ ≤i n

( )

2 11 3, 1 i

[image:5.595.237.394.84.237.2]

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[image:6.595.145.505.83.714.2] [image:6.595.152.484.84.311.2]

Figure 7. The labeling pattern of 4DS P

(

2K1

)

.

Figure 8. The graph DS P

(

₂K₂

)

.

( )

3 11 2, 1 i

f v = i− ≤ ≤i n

( )

4 11 1, 1 i

f v = i− ≤ ≤i n

( )

5 11 9, 1 i

f v = i− ≤ ≤i n

( )

6 11 7, 1 i

f v = i− ≤ ≤i n

( )

1 11 , 1 i

f w = i ≤ ≤i n

( )

2 11 10, 1 i

f w = i− ≤ ≤i n

( )

5 6 11 8, 1 i i

f v v = i− ≤ ≤i n

( )

5 1 11 7, 1 i i

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( )

5 2 11 6, 1 i i

f v v = i− ≤ ≤i n

( )

6 3 11 5, 1 i i

f v v = i− ≤ ≤i n

( )

6 4 11 4, 1 i i

f v v = i− ≤ ≤i n

( )

1 1 11 3, 1 i i

f v w = i− ≤ ≤i n

( )

2 1 11 2, 1 i i

f v w = i− ≤ ≤i n

( )

3 1 11 1, 1 i i

f v w = i− ≤ ≤i n

( )

4 1 11 , 1 i i

f v w = i ≤ ≤i n

( )

5 2 11 10, 1 i i

f v w = i− ≤ ≤i n

( )

6 2 11 9, 1 i i

f v w = i− ≤ ≤i n

{

1, 2,,q

}

(

₂K₂

)

is shown inFigure 9. Theorem 2.9: nDS P

(

₂K₃

)

(

₂K₃

)

Let G=nDS P

(

₂K₃

)

{

1, 2, 3, 4, 5, 6, 7, 8, 1, 2,1

}

i i i i i i i i i i i

V = v v v v v v v v w w ≤ ≤i n .

Define a function f V G:

( ) {

→ 1, 2,,q+1

}

by

( )

1 15 11, 1 i

f v = i− ≤ ≤i n

( )

2 15 8, 1 i

f v = i− ≤ ≤i n

( )

3 15 6, 1 i

f v = i− ≤ ≤i n

( )

4 15 5, 1 i

f v = i− ≤ ≤i n

( )

5 15 3, 1 i

f v = i− ≤ ≤i n

( )

6 15 2, 1 i

f v = i− ≤ ≤i n

( )

1 15 , 1 i

f w = i ≤ ≤i n

( )

2 15 14, 1 i

f w = i− ≤ ≤i n

( )

7 15 13, 1 i

f v = i− ≤ ≤i n

( )

8 15 12, 1 i

f v = i− ≤ ≤i n

( )

7 1 15 11, 1 i i

f v v = i− ≤ ≤i n

( )

7 2 15 10, 1 i i

f v v = i− ≤ ≤i n

( )

7 3 15 9, 1 i i

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[image:8.595.104.531.80.716.2]

Figure 9.The labeling pattern of 2DS P

(

₂K₂

)

.

Figure 10. The graph DS P

(

2K3

)

.

( )

8 4 15 8, 1 i i

f v v = i− ≤ ≤i n

( )

8 5 15 7, 1 i i

f v v = i− ≤ ≤i n

( )

8 6 15 6, 1 i i

f v v = i− ≤ ≤i n

( )

7 2 15 14, 1 i i

f v w = i− ≤ ≤i n

( )

8 2 15 13, 1 i i

f v w = i− ≤ ≤i n

( )

1 1 15 5, 1 i i

f v w = i− ≤ ≤i n

( )

2 1 15 4, 1 i i

f v w = i− ≤ ≤i n

( )

3 1 15 3, 1 i i

f v w = i− ≤ ≤i n

( )

4 1 15 2, 1 i i

f v w = i− ≤ ≤i n

( )

5 1 15 1, 1 i i

f v w = i− ≤ ≤i n

( )

6 1 15 , 1 i i

f v w = i ≤ ≤i n

( )

7 8 15 12, 1 i i

[image:8.595.105.523.83.237.2]

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{

1, 2,,q

}

Example 2.10: The root square mean labeling of 2DS P

(

₂K₃

)

Theorem 2.11:nDS P

(

₃K₁

)

is a root square mean graph. Proof: The graph DS P

(

₃K₁

)

Let G=nDS P

(

₃K₁

)

. Let its vertex set be V =V₁  V₂  V_n

where Vi =

{

v v v v v v w w₁i, ₂i, ₃i, ₄i, ₅i, ₆i, ₁i, ₂i,1≤ ≤i n

}

.

( ) {

→ 1, 2,,q+1

}

by

( )

1 10 3, 1 i

f v = i− ≤ ≤i n

( )

2 10 2, 1 i

f v = i− ≤ ≤i n

( )

3 10 1, 1 i

f v = i− ≤ ≤i n

( )

4 10 8, 1 i

f v = i− ≤ ≤i n

( )

5 10 5, 1 i

f v = i− ≤ ≤i n

( )

6 10 6, 1 i

f v = i− ≤ ≤i n

( )

1 10 , 1 i

f w = i ≤ ≤i n

( )

2 10 9, 1

i

[image:9.595.100.523.92.679.2]

f w = i− ≤ ≤i n

Figure 11. The root square mean labeling of 2DS P

(

₂K₃

)

.

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( )

4 1 10 5, 1 i i

f v v = i− ≤ ≤i n

( )

5 2 10 4, 1 i i

f v v = i− ≤ ≤i n

( )

6 3 10 3, 1 i i

f v v = i− ≤ ≤i n

( )

1 1 10 2, 1 i i

f v w = i− ≤ ≤i n

( )

2 1 10 1, 1 i i

f v w = i− ≤ ≤i n

( )

3 1 10 , 1 i i

f v w = i ≤ ≤i n

( )

4 2 10 9, 1

i i

f v w = i− ≤ ≤i n

( )

6 2 10 8, 1 i i

f v w = i− ≤ ≤i n

{

1, 2,,q

}

(

₃K₁

)

is shown inFigure 13. Theorem 2.13: nDS K

( )

_1,3 is a root square mean graph.

[image:10.595.137.493.312.726.2]

Proof: The graph DS K

( )

_1,3 is shown inFigure 14.

Figure 13. The labeling pattern of 3DS P

(

₃K₁

)

.

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Let G=nDS K

( )

_1,3 . Let its vertex set be V =V₁  V₂  V_n where Vi=

{

v v v v w₁i, ₂i, ₃i, ₄i, i,1≤ ≤i n

}

.

( ) {

→ 1, 2,,q+1

}

by

( )

1 6 5, 1 i

f v = −i ≤ ≤i n

( )

2 6 4, 1 i

f v = −i ≤ ≤i n

( )

3 6 2, 1 i

f v = −i ≤ ≤i n

( )

4 6 1, 1 i

f v = −i ≤ ≤i n

( )

i 6 , 1 f w = i ≤ ≤i n

( )

1 2 6 5, 1 i i

f v v = −i ≤ ≤i n

( )

1 3 6 4, 1 i i

f v v = −i ≤ ≤i n

( )

1 4 6 3, 1 i i

f v v = −i ≤ ≤i n

( )

2 6 2, 1

i i

f v w = −i ≤ ≤i n

( )

3 6 1, 1

i i

f v w = −i ≤ ≤i n

( )

4 6 , 1 i i

f v w = i ≤ ≤i n

{

1, 2,,q

}

Example 2.14: The labeling pattern of 4DS K

( )

_1,3 is shown inFigure 15. Theorem 2.15: nDS C

(

₃OˆK_1,2

)

Proof: The graph DS C

(

₃OˆK_1,2

)

[image:11.595.133.481.489.709.2]

Let G=nDS C

(

₃OˆK_1,2

)

. Let its vertex set be V =V₁  V₂  V_n

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[image:12.595.79.538.80.677.2]

Figure 16.The graph DS C

(

3OˆK1,2

)

.

where Vi=

{

v v v v v w w₁i, ₂i, ₃i, ₄i, ₅i, ₁i, ₂i,1≤ ≤i n

}

.

( ) {

→ 1, 2,,q+1

}

by

( )

1 9 7, 1 i

f v = −i ≤ ≤i n

( )

2 9 5, 1 i

f v = −i ≤ ≤i n

( )

3 9 4, 1 i

f v = −i ≤ ≤i n

( )

4 9 2, 1 i

f v = −i ≤ ≤i n

( )

5 9 1, 1 i

f v = −i ≤ ≤i n

( )

1 9 8, 1

i

f w = −i ≤ ≤i n

( )

2 9 , 1 i

f w = i ≤ ≤i n

( )

1 1 9 8, 1 i i

f w v = −i ≤ ≤i n

( )

1 2 9 7, 1

i i

f w v = −i ≤ ≤i n

( )

1 2 9 6, 1 i i

f v v = −i ≤ ≤i n

( )

1 3 9 5, 1 i i

f v v = −i ≤ ≤i n

( )

2 3 9 4, 1 i i

f v v = −i ≤ ≤i n

( )

3 4 9 3, 1 i i

f v v = −i ≤ ≤i n

( )

3 5 9 2, 1 i i

f v v = −i ≤ ≤i n

( )

4 2 9 1, 1 i i

f v w = −i ≤ ≤i n

( )

5 2 9 , 1 i i

f v w = i ≤ ≤i n

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[image:13.595.146.501.78.290.2]

Figure 17.The root square mean labeling of 4DS C

(

₃OˆK_1,2

)

.

Example 2.16: The root square mean labeling of 4DS C

(

₃OˆK_1,2

)

is given inFigure 17.

References

[1] Gallian, J.A. (2012) A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatories. [2] Harary, F. (1988) Graph Theory. Narosa Publishing House Reading, New Delhi.

[3] Sandhya, S.S., Jayasekaran, C. and Raj, C.D. (2013) Harmonic Mean Labeling of Degree Splitting Graphs. Bulletin of Pure and Applied Sciences, 32E, 99-112.

[4] Sandhya, S.S., Somasundaram, S. and Anusa, S. (2014) Root Square Mean Labeling of Graphs. International Journal of Contemporary Mathematical Sciences, 9, 667-676.

[5] Sandhya, S.S., Somasundaram, S. and Anusa, S. (2015) Some More Results on Root Square Mean Graphs. Journal of Mathematics Research, 7.

[6] Sandhya, S.S., Somasundaram, S. and Anusa, S. (2014) Root Square Mean Labeling of Some New Disconnected Graphs. International Journal of Mathematics Trends and Technology, 15, 85-92.

http://dx.doi.org/10.14445/22315373/IJMTT-V15P511

[7] Sandhya, S.S., Somasundaram, S. and Anusa, S. (2014) Root Square Mean Labeling of Subdivision of Some More Graphs. International Journal of Mathematics Research, 6, 253-266.

[8] Sandhya, S.S., Somasundaram, S. and Anusa, S. (2014) Some New Results on Root Square Mean Labeling. Interna-tional Journal of Mathematical Archive, 5, 130-135.

[9] Sandhya, S.S., Somasundaram, S. and Anusa, S. (2015) Root Square Mean Labeling of Subdivision of Some Graphs.

Global Journal of Theoretical and Applied Mathematics Sciences, 5, 1-11.

[10] Sandhya, S.S., Somasundaram, S. and Anusa, S. (2015) Root Square Mean Labeling of Some More Disconnected Graphs. International Mathematical Forum, 10, 25-34.