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Journal of Mathematics Research; Vol. 7, No. 1; 2015
ISSN 1916-9795 E-ISSN 1916-9809
Published by Canadian Center of Science and Education
72
Some More Results on Root Square Mean Graphs
S. S. Sandhya1, S. Somasundaram2 & S. Anusa3
1Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai 629003, India.
2Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627012, India.
3Department of Mathematics, Arunachala College of Engineering for Women, Vellichanthai 629203, India.
Correspondence: S. Anusa, Department of Mathematics, Arunachala College of Engineering for Women, Vellichanthai
629203, India. E-mail: anu12343s@gmail.com
Received: October 23, 2014 Accepted: November 6, 2014 Online Published: February 17, 2015
doi:10.5539/jmr.v7n1p72 URL: http://dx.doi.org/10.5539/jmr.v7n1p72
Abstract
A graph with vertices and edges is called a Root Square Mean graph if it is possible to label
the vertices with distinct elements from in such a way that when each edge
is labeled with
or
, then the resulting edge labels are distinct. In
this case is called a Root Square Mean labeling of . The concept of Root Square Mean labeling was
introduced by (S. S. Sandhya, S. Somasundaram and S. Anusa). We investigated the Root Square Mean labeling
of several standard graphs such as Path, Cycle, Comb, Ladder, Triangular snake, Quadrilateral snake etc., In this
paper, we investigate the Root Square Mean labeling for Double Triangular snake, Alternate Double Triangular
snake, Double Quadrilateral snake, Alternate Double Quadrilateral snake, and Polygonal chain.
Keywords: Mean graph, Root Square Mean graph, Cycle, Triangular snake, Double Triangular snake,
Quadrilateral snake, Double Quadrilateral snake, Polygonal chain.
1. Introduction
The graph considered here will be finite, undirected and simple. The vertex set is denoted by and the edge
set is denoted by .For all detailed survey of graph labeling we refer to Gallian (2010). For all other
standard terminology and notations we follow Harary (1988). A Triangular snake is obtained from a path
by joining and to a new vertex for . A Double Triangular Snake
consists of two Triangular snakes that have a common path. An Alternate Triangular snake is
obtained from a path by joining and (Alternatively) to new vertex .An Alternate
Double Triangular Snake consists of two Alternate Triangular snakes that have a common path. A
Quadrilateral snake is obtained from a path by joining and to new vertices and
respectively and then joining and. A Double Quadrilateral snake consists of two Quadrilateral
snakes that have a common path. An Alternate Quadrilateral snake is obtained from a path
by joining and (Alternatively) to new vertices and respectively and then joining and.An
Alternate Double Quadrilateral snake consists of two Alternate Quadrilateral snakes that have a
common path.A Polygonal chain is a connected graph all of whose blocks are polygons on sides.
S. Somasundaram and R. Ponraj introduced the concept of mean labeling of graphs and investigated the mean
labeling of some standard graphs. S. Somasundaram and S. S. Sandhya introduced the concept of Harmonic
mean labeling of graphs. S. Somasundaram and P. Vidhya Rani introduced the concept of Geometric mean
labeling of graphs. In this paper we prove that Double Triangular snake, Alternate Double Triangular snake,
Double Quadrilateral snake, Alternate Double Quadrilateral snake and Polygonal Chains are Root Square mean
graphs.
We make frequent reference to the following results.
Theorem 1.1: (S. Somasundaram & R. Ponraj) Triangular snakes and Quadrilateral snakes are mean graphs.
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Theorem 1.2: (S. S. Sandhya, S. Somasundaram & S. Anusa) Double Triangular and Double Quadrilateral
snakes are mean graphs.
Theorem 1.3: (S. S. Sandhya & S. Somasundaram) Triangular snakes and Quadrilateral snakes are Harmonic
mean graphs.
Theorem 1.4: (C. Jaya Sekaran, S. S. Sandhya & C. David Raj) Double Triangular snakes and Alternate Double
Triangular snakes are Harmonic mean graphs.
Theorem 1.5: (C. David Raj, C. Jaya Sekaran & S. S. Sandhya) Double Quadrilateral and Alternate Double
Quadrilateral snakes are Harmonic mean graphs.
Theorem 1.6: (S. S. Sandhya & S. Somasundaram) Double Triangular snakes are Geometric mean graphs.
Theorem 1.7: (S. S. Sandhya & S. Somasundaram) Double Quadrilateral snakes are Geometric mean graphs.
2. Root Square Mean Labeling
Definition 2.1: A graph with vertices and edges is called a mean graph if it is possible to
label the vertices with distinct elements from in such a way that when each edge
is labeled with
, then the edge labels are distinct.
In this case is called Mean labeling of .
Definition 2.2: A graph with vertices and edges is called as Harmonic mean graph if it is possible to label
the vertices with distinct elements from in such a way that when each edge
is labeled with
, then the edge labels are distinct. In this case
is called Harmonic mean labeling of .
Definition 2.3: A graph with vertices and edges is called as Geometric mean graph if it is possible to label
the vertices with distinct elements from in such a way that when each edge
is labeled with , then the edge labels are distinct. In this
case is called Geometric mean labeling of .
Definition 2.4: A graph with vertices and edges is called a Root Square mean graph if it is possible to
label the vertices with distinct elements from in such a way that when each edge
is labeled with
, then the edge labels are distinct. In
this case is called Root Square mean labeling of .
Theorem 2.5: Double Triangular snakes are Root square mean graphs.
Proof: Consider a path .Join and , to two new vertices , ,
.Define a function by
,
.
The edges are labeled as
Then the edge labels are distinct. Hence Double Triangular snakes are Root Square mean graphs.
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Example 2.6: Root Square mean labeling of is given below.
Figure 1
Theorem 2.7: Alternate Double Triangular snakes is a Root Square mean graph.
Proof: Let be the graph . Consider a path .To construct , join and
(Alternatively) with two new vertices and , .There are two different cases to be considered.
Case 1: If the Double Triangle starts from , then we consider two sub cases.
Sub Case 1(a): If is even, then
Define a function by
,
.
The edges are labeled as
Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 2
Sub Case 1(b): If is odd then
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Define a function by
,
.
The edges are labeled as
Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 3
Case 2: If the triangle starts from , then we have to consider two sub cases.
Sub case 2(a): If is even , then
Define a function by
,
.
The edges are labeled as
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Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 4
Sub Case 2(b): If is odd
Define a function by
,
.
The edges are labeled as
Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 5
From all the above cases, we conclude that Alternate Double Triangular Snakes are Root Square
mean graphs.
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Theorem 2.8: Double Quadrilateral snake graph is a Root Square mean graphs.
Proof: Let be the path .To construct, join and to four new vertices ′
and ′ by the edges ′′ and ′′ , for
Define a function by
,
,
′′,
′′.
The edges are labeled as
,
′ ′
′,
′′
Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
Example 2.9: The Root Square mean labeling of is given below.
Figure 6
Theorem 2.10: Alternate Double Quadrilateral snake graphs are Root Square mean graphs.
Proof: Let be the Alternate Double Quadrilateral snake .Consider a path .Join
and (Alternatively) with to four new vertices and .Here we consider two different
cases.
Case 1: If the Double Quadrilateral starts from , then we consider two sub cases.
Sub Case 1(a): If is even then
Define a function by
.
The edges are labeled as
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Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 7
Sub Case 1(b): If is odd then
Define a function by
,
.
The edges are labeled as
Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 8
Case 2: If the Double Quadrilateral starts from , then we have to consider two sub cases.
Sub case 2(a): If is even , then
Define a function by
,
.
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The edges are labeled as
Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 9
Sub Case 2(b): If is odd , then
Define a function by
.
The edges are labeled as
Then the edge labels are distinct. Hence in this case is a Root Square mean labeling of .
The labeling pattern is shown below.
Figure 10
Theorem 2.11: Polygonal chain are Root Square mean graphs for all and .
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Proof: In , let be the first cycle. The
second cycle is connected to the first cycle at the vertex .
Let be the second cycle.In general the rth cycle is
connected to the (r-1)th cycle at the vertex .Let the rth cycle be
.The figure of the rth cycle is given below.
Figure 11
Let the graph has cycles . Define a function by
, . Then the label of the edges is given below.
, , ,
.
Hence the graph has distinct edge labels, hence is a Root Square mean graph.
Example 2.12: Root Square mean labeling of chain is given below.
Figure 12
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