On weighted noncorona graphs with property R and −SR

10.48129/kjs.17497

Authors

  • Uzma Ahmad University of the Punjab, Lahore, Pakistan
  • Saira Hameed University of the Punjab
  • Sadia Akhter University of the Punjab

DOI:

https://doi.org/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.

Author Biographies

Uzma Ahmad, University of the Punjab, Lahore, Pakistan

Assistant Professor, Department of Mathematics

Saira Hameed, University of the Punjab

Assitant Professor,  Department of Mathematics, University of the Punjab

Sadia Akhter, University of the Punjab

Department of Mathematics, University of the Punjab

Published

06-07-2022

Issue

Section

Mathematics