On weighted noncorona graphs with properties R and −SR

10.48129/kjs.17497

Abstract

Let $\mathfrak{G}_w$ be a simple undirected weighted graph with adjacency matrix $\mathcal{A}(\mathfrak{G}_w)$. The set of all eigenvalues of $\mathfrak{G}_w$ is called the spectrum of $\mathfrak{G}_w$ denoted by $\sigma(\mathfrak{G}_w)$. The reciprocal eigenvalue property (or property $\mathcal{R})$ for a nonsingular graph
$\mathfrak{G}$ is defined as, if $\eta \in \sigma(\mathfrak{G})$ then $\frac{1}{\eta}\in \sigma(\mathfrak{G})$. Further, if $\eta$ and $\frac{1}{\eta}$ have
same multiplicities $\forall ~\eta\in \sigma(\mathfrak{G})$ then graph is said to have strong reciprocal
eigenvalue property (or property $\mathcal{SR}$. Similarly, a nonsingular graph G
is said to have anti­reciprocal eigenvalue property (or property $-\mathcal{R}$) if $\eta \in \sigma(\mathfrak{G})$ then $-\frac{1}{\eta}\in \sigma(\mathfrak{G})$ furthermore if $\eta$ and $-\frac{1}{\eta}$ have same multiplicities $\forall ~\eta\in \sigma(\mathfrak{G})$ then strong anti­reciprocal eigenvalue
property (or property $-\mathcal{SR}$) holds for graph. In this article, classes of weighted noncorona graphs satisfying property $\mathcal{R}$ and property $-\mathcal{SR}$ are studied.