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Copyrights @Kalahari Journals Vol. 6 No. 3(December, 2021)
International Journal of Mechanical Engineering
3007
ISSN: 0974-5823 Vol. 6 No. 3 December, 2021
International Journal of Mechanical Engineering
A STUDY ON STRONGLY REGULAR GRAPHS
C.J. Chris Lettecia Mary1, R. Kamali 2
1Research Scholar, Department of Mathematics,
Vels Institute of Science, Technology and Advanced Studies,
Chennai-600 117, Tamil Nadu, India.
2 Assistant Professor, Department of Mathematics,
Vels Institute of Science, Technology and Advanced Studies,
Chennai-600 117, Tamil Nadu, India.
Abstract
In this paper, the graphs which were regular are discussed to
be strongly regular graphs. Also certain theorems were proved
based on strongly regular graphs with suitable examples.
Keywords: Cycle, Eigen values, Graph, Strongly regular
Graph.
1. Introduction:
In Mathematics field, Graph theory is the learning of Graphs
that means the tie between points and lines as vertices and
edges. A Graph is a graphic description of a group of things
where pair of things are merged by links. Graph Theory are
also applied in Computer Science, Electrical Engineering,
Physics and Chemistry.
Algebraic Graph Theory is a branch of Mathematics that
involves to find the solutions for Algebraic Methods by using
Graph Theory concepts. Linear Algebra, Group Theory and
the study of Graph invariants are three main divisions of
Algebraic Graph Theory. There is a bind between Graph
Theory and Group Theory, which is shown by Arthur Cayley.
He was the first to introduce the Cayley Graphs to finite
groups.
We define Graph as a pair , where V is a non-empty set
and E is the set of edges. A Graph is called Regular Graph if
all the vertices having the same number of degrees. 0-regular
graph is a empty or null graph. 1-regular graph is always a
disconnected Graph. 3-regular graphs are also called Cubic
Graph.
The main aim of this paper is to study on strongly regular
graphs. Also we derived some theorems on Strongly Regular
Graphs with suitable examples.
2. Preliminaries
Definition 2.1
A Graph consists of a vertex set and an edge set
, where edge is an unordered pair of distinct vertices of
.
Definition 2.2
A graph in which degree of all the vertices are same is called
as regular graph. If all the vertices in a graph are of degree ,
then it is called k-regular graph.
Definition 2.3
An undirected graph is a graph without any direction of
edges.
Example 2.1:
Figure 2.1 – Undirected Graph
In the Figure 2.1, the vertices are 1,2,3,4. (1,2), (1,4),
(2,3), (2,4) are the edges of the graph. Since it is undirected
graph the edges (1,2) and (2,1) are same. Similarly other edges
are also considered in the same way.
Definition 2.4
The eccentricity of a vertex in is the maximum
distance from to any other vertex .
Definition 2.5
The diameter of a graph is denoted by and is
defined by
Definition 2.6
Let be a graph with ,
and without parallel edges. The adjacency
Copyrights @Kalahari Journals Vol. 6 No. 3(December, 2021)
International Journal of Mechanical Engineering
3008
matrix of is an symmetric binary defined
as integers such that
Example 2.2:
Consider the graph
Figure 2.2 – Graph X
The adjacency matrix of X is given by
A=
Example 2.3:
The adjacency matrix of the Petersen graph of order 10
represented in Figure 3.1 is as follows:
B =
Definition 2.7
The graph on vertices is strongly regular with
parameters if
i) is k-regular, such that every vertex in has k
neighbours.
ii) Each pair of adjacent vertices has exactly common
neighbours.
iii) Each pair of non-adjacent vertices has exactly common
neighbours.
Strongly regular graphs are denoted by .
Example 2.4:
The cycle graph with 5 vertices is represented in figure 3
Figure 2.3 – Cycle
Here, is (5, 2, 0, 1)
Definition 2.8
An Eigen vector of a matrix is a non-zero vector
such that for some scalar . The scalar is the Eigen
value of .
3. Strongly Regular Graphs
Theorem 3.1
If then
Proof:
Given is a strongly regular graph with vertices with
neighbours
Let be a vertex in
There are two ways to find the number of edges between the
neighbours and non-neighbours of .
Case: (i)
u has a set of neighbours.
Let neighbours of
Where which is adjacent to
And also is adjacent to the non-neighbours of .
There are non neighbours and there are
neighbours of .
Therefore, non neighbours of u.
Case: (ii)
Every non neighbours of is adjacent to .
Therefore, non neighbours of .
Hence .
Copyrights @Kalahari Journals Vol. 6 No. 3(December, 2021)
International Journal of Mechanical Engineering
3009
Example 3.1:
The Petersen graph with 10 vertices is represented in the
following figure
Figure 3.1 – Petersen Graph
Here, = (10, 3, 0,1)
Remark 3.1:
If is strongly regular, then
is also strongly regular and
=
Example 3.2:
Let us take Petersen graph, which is strongly regular graph
and
= (10, 3, 0, 1)
The complement of Petersen graph is represented in Figure
3.2
Figure 3.2 – Complement of Petersen Graph
Here, = (10, 6, 3, 4)
Note 3.1 :
Strongly regular graph is disconnected if and only if
Theorem 3.2
Every connected strongly regular graph has diameter 2.
Proof:
Given is a strongly regular graph with vertices with
neighbours
Let u be a vertex in
is adjacent with some where
And is not adjacent with some i where
But is adjacent with
The eccentricity of is 2 [By Definition 2.4]
Therefore, the diameter of any strongly regular graph is two.
Remark 3.2:
Let be a strongly regular graph with parameters .
Then the numbers
=
are non-negative
integers.
Example 3.3:
Let us take the complement of the Petersen graph,
= (10, 6, 3, 4)
Then by the above remark 3.2 we get
=
and =
Here, are non-negative integers.
Theorem 3.3
Let be a strongly regular graph with parameters
and Eigen values , and . Then the complement of (
has eigenvalues (n-k-1), (-r-1) and (–s-1). Moreover the Eigen
spaces of and X are same.
Proof:
Let B be the adjacency matrix of X.
Then the adjacency matrix of is =
We know that, is an Eigen value of and it’s Eigen space is
the space of constant vectors.
Let be a constant vector.
We have =
=
=
Therefore, is an eigen value of
And the eigen space is same as the eigen space of .
Let be an eigen vector of corresponding to the eigen
vector .
We know that, is orthogonal to .
y =
=
=
Copyrights @Kalahari Journals Vol. 6 No. 3(December, 2021)
International Journal of Mechanical Engineering
3010
Therefore, is an eigen value of .
And it’s eigen space is as same as the eigen space of .
Similarly, is an eigen value of .
Hence the proof.
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