On graphs with anti-reciprocal eigenvalue property

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Akhter Sadia; Ahmad Uzma; Hameed Saira

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Journal Paper

Paper Information

Journal: TRANSACTIONS ON COMBINATORICS Year:2024 | Volume:13 | Issue:1 Page(s): 17-30

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Information Journal Paper

Title

On graphs with anti-reciprocal eigenvalue property

Author(s)

Akhter Sadia | Ahmad Uzma | Hameed Saira | Issue Writer Certificate

Keywords

Anti-reciprocal eigenvalue property

strong anti-reciprocal eigenvalue property

Adjacency Matrix

graph spectrum

Abstract

Let $\mathtt{A}(\mathtt{G})$ be the Adjacency Matrix of a simple connected undirected graph $\mathtt{G}$. A graph $\mathtt{G}$ of order $n$ is said to be non-singular (respectively singular) if $\mathtt{A}(\mathtt{G})$ is non-singular (respectively singular). The spectrum of a graph $\mathtt{G}$ is the set of all its eigenvalues denoted by $spec(\mathtt{G})$. The anti-reciprocal (respectively reciprocal) eigenvalue property for a graph $\mathtt{G}$ can be defined as `` Let $\mathtt{G}$ be a non-singular graph $\mathtt{G}$ if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of $\mathtt{G}$, then $\mathtt{G}$ has anti-reciprocal (respectively reciprocal) eigenvalue property ." Furthermore, a graph $\mathtt{G}$ is said to have strong Anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property $(-\mathtt{R})$) and strong Anti-reciprocal eigenvalue property (or property $(-\mathtt{SR})$) are discussed.

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