On spectrum of the zero-divisor graph of matrix ring

TL;DR

This paper investigates the spectral properties of the zero-divisor graph of the matrix ring over a finite field, providing bounds on eigenvalues using Weyl's inequality.

Contribution

It introduces bounds on the eigenvalues of the adjacency matrix of the zero-divisor graph of 2x2 matrix rings over finite fields using Weyl's inequality.

Findings

Eigenvalue bounds for the adjacency matrix of the zero-divisor graph

Application of Weyl's inequality to matrix ring graphs

Insights into spectral properties of matrix ring graphs

Abstract

For a ring $R$, the zero-divisor graph is a simple graph $\Gamma(R)$ whose vertex set is the set of all non-zero zero-divisors in a ring $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$ or $yx=0$ in $R$. By using Weyl's inequality we give bounds on eigenvalues of adjacency matrix of $\Gamma(M_2(F))$, where $M_2(F)$ is a $2 \times 2$ matrix ring over a finite field $F$.