Seidel energy of a graph with self-loops

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India, 576104

2 Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia

Abstract

Let $G_S$ be a graph obtained by attaching a self-loop to each vertex of $S\subseteq V$  of a graph $G(V,E)$. The Seidel matrix of $G_S$ is $S(G_S)=[s_{ij}]$, where $s_{ij}=-1$ if $v_i$ and $v_j$ are adjacent and $v_i\in S$, $s_{ij}=1$ if $v_i$ and $v_j$ are non-adjacent, and it is zero if $i=j$ and $v_i\not\in S$.     If $\theta_i(G_S)\,,\,i=1,2,\ldots,n$, are the eigenvalues of the Seidel matrix, then the Seidel energy of the graph $G_S$, containing $n$ vertices and $\sigma$ self-loops, is defined as $\sum_{i=1}^n \left|\theta_i(G_S)+\frac{\sigma}{n}\right|$. In this paper, some basic properties of Seidel energy of graphs containing self-loops are established.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 13 September 2024

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