Applied Mathematics
Vol.06 No.06(2015), Article ID:56857,12 pages
10.4236/am.2015.66086
Degree Splitting of Root Square Mean Graphs
S. S. Sandhya1, S. Somasundaram2, S. Anusa3
1Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai, India
2Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, India
3Department of Mathematics, Arunachala College of Engineering for Women, Vellichanthai, India
Email: anu12343s@gmail.com
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 15 April 2015; accepted 30 May 2015; published 2 June 2015
ABSTRACT
Let 
be an injective function. For a vertex labeling f, the induced edge labeling 
is defined by, 
or 
then, the edge labels are distinct and are from
. 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 
denote the vertex set and 
denote the edge set of G. For detailed survey of graph labeling we refer to Gallian [1] . For all other standard terminology and notations we follow Harary [2] . The concept of mean labeling on degree splitting graph was introduced 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.
Definition 1.1: A graph 
with p vertices and q edge is called a root square mean graph if it is possible to label the vertices 
with distinct labels 
from 
in such a way that when
each edge 
is labeled with 
or
, then the edge
labels are distinct and are from
. In this case f is called root square mean labeling of G.
Definition 1.2: A walk in which 
are distinct is called a path. A path on n vertices is denoted by
.
Definition 1.3: A closed path is called a cycle. A cycle on n vertices is denoted by
.
Definition 1.4: Let 
be a graph with
, where each 
is a set of vertices having at least two vertices and having the same degree and
. The degree splitting graph of G is denoted by 
and is obtained from G by adding the vertices 
and joining 
to each vertex of 
The graph G and its degree splitting graph 
are given in Figure 1.
Definition 1.5: The union of two graphs 
and 
is a graph 
with vertex set 
and the edge set
.
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: 
is a root square mean graph.
Proof: The graph 
is shown in Figure 2.
Let
. Let the vertex set of G be 
where
. Define a function 
by
Figure 1. The graph G and its degree splitting graph
.
Figure 2. The graph
.
Then the edges are labeled as
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.2: Root square mean labeling of 
is shown in Figure 3.
Theorem 2.3:
is a root square mean graph.
Proof: The graph 
is shown in Figure 4.
Let
. Let the vertex set of G be 
where
. Define a function 
by
Figure 3. Root square mean labeling of
.
Figure 4. The graph
.
Then the edges are labeled as
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.4: Root square mean labeling of 
is shown in Figure 5.
Theorem 2.5: 
is a root square mean graph.
Proof: The graph 
is shown in Figure 6.
Let
. Let the vertex set of G be 
where
. Define a function 
by
Figure 5. Root square mean labeling of
.
Figure 6. The graph
.
Then the edges are labeled as
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.6: The labeling pattern of 
is shown in Figure 7.
Theorem 2.7:
is a root square mean graph.
Proof: The graph 
is shown in Figure 8.
Let
. Let the vertex set of G be 
where
. Define a function 
by
Figure 7. The labeling pattern of
.
Figure 8. The graph
.
Then the edges are labeled as
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.8: The labeling pattern of 
is shown in Figure 9.
Theorem 2.9: 
is a root square mean graph.
Proof: The graph 
is shown in Figure 10.
Let
. Let the vertex set of G be 
where
.
Define a function 
by
Then the edges are labeled as
Figure 9. The labeling pattern of
.
Figure 10. The graph
.
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.10: The root square mean labeling of 
is shown in Figure 11.
Theorem 2.11:
is a root square mean graph.
Proof: The graph 
is shown in Figure 12.
Let
. Let its vertex set be 
where
.
Define a function 
by
Figure 11. The root square mean labeling of
.
Figure 12. The graph
.
Then the edges are labeled as
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.12: The labeling pattern of 
is shown in Figure 13.
Theorem 2.13: 
is a root square mean graph.
Proof: The graph 
is shown in Figure 14.
Figure 13. The labeling pattern of
.
Figure 14. The graph
.
Let
. Let its vertex set be 
where
.
Define a function 
by
Then the edges are labeled as
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.14: The labeling pattern of 
is shown in Figure 15.
Theorem 2.15: 
is a root square mean graph.
Proof: The graph 
is shown in Figure 16.
Let
. Let its vertex set be 
Figure 15. The labeling pattern of
.
Figure 16. The graph
.
where
.
Define a function 
by
Then the edges are labeled as
Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Figure 17. The root square mean labeling of
.
Example 2.16: The root square mean labeling of 
is given in Figure 17.
References
- Gallian, J.A. (2012) A Dynamic Survey of Graph Labeling. The Electronic Journal of Combinatories.
- Harary, F. (1988) Graph Theory. Narosa Publishing House Reading, New Delhi.
- 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.
- Sandhya, S.S., Somasundaram, S. and Anusa, S. (2014) Root Square Mean Labeling of Graphs. International Journal of Contemporary Mathematical Sciences, 9, 667-676.
- Sandhya, S.S., Somasundaram, S. and Anusa, S. (2015) Some More Results on Root Square Mean Graphs. Journal of Mathematics Research, 7.
- 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
- 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.
- Sandhya, S.S., Somasundaram, S. and Anusa, S. (2014) Some New Results on Root Square Mean Labeling. International Journal of Mathematical Archive, 5, 130-135.
- 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.
- Sandhya, S.S., Somasundaram, S. and Anusa, S. (2015) Root Square Mean Labeling of Some More Disconnected Graphs. International Mathematical Forum, 10, 25-34.
- Sandhya, S.S., Somasundaram, S. and Anusa, S. (2015) Some Results on Root Square Mean Graphs. Communicated to Journal of Scientific Research.