Article
KYUNGPOOK Math. J. 2019; 59(2): 335-340
Published online June 23, 2019
Copyright © Kyungpook Mathematical Journal.
Some New Results on Seidel Equienergetic Graphs
Samir K. Vaidya∗, Kalpesh M. Popat
Department of Mathematics, Saurashtra University, Rajkot-360005, Gujarat, India
e-mail : samirkvaidya@yahoo.co.in
Department of Master of Computer Application, Atmiya Institute Of Technology & Science, Rajkot-360005, Gujarat, India
e-mail : kalpeshmpopat@gmail.com
Received: March 30, 2018; Revised: April 27, 2019; Accepted: May 10, 2019
Abstract
The energy of a graph
1. Introduction
For standard terminology and notations related to graph theory we follow Balakrishnan and Ranganathan [2] while for any undefined term in algebra we follow Lang [10]. Let
The eigenvalues of
The other varients of energy like Laplacian energy [7], Incidence energy [6], Skew energy [1], Distance energy [3], Seidel energy [8] are also available in the literature. In the present paper we have focused on Seidel energy of graphs.
Let
The eigenvalues of the Seidel matrix, labeled as
As an example the, Seidel matrix of the complete graph
where a power denotes the multiplicity of an eigenvalue. Therefore,
Two graphs
The
Ramane
2. Seidel Equienergetic Graphs
To state our results we need to defined some graphs. The
Let
It is well known and easy to show that for graphs
The following is useful for determining the spectrum of a product of graphs from those of its factors.
Proposition 2.1.([9])
Let
and
We introduce two convenient notations. Let
Lemma 2.2
As
and so taking
Lemma 2.3
Much like above,
Theorem 2.4
It is not hard to see that the sums of the spectrums in these two lemma are the same. To say the same about the sums of the absolute values, (so that the Seidel energies are the same) we observe that we need that
Assuming that
Observe that if
Theorem 2.5
By Lemma 2.2,
If
3. Concluding Remarks
The concept of Seidel equienergetic graphs is analogous to the concepts of equienergetic graphs. We present here methods to construct Seidel equienergetic graphs by means of
Acknowledgements
The authors are highly thankful to the anonymous referee for kind suggestions and comments on the first draft of this paper.
Figures
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