Spectra of the zero-divisor graph of finite rings

TL;DR

This paper investigates the spectral properties of the zero-divisor graph of finite rings, providing a new structural decomposition and explicit spectra for finite semisimple rings.

Contribution

It introduces an equivalence relation on rings to express zero-divisor graphs as generalized joins and computes their spectra for finite semisimple rings.

Findings

Spectra of zero-divisor graphs are characterized for finite semisimple rings.

Zero-divisor graphs can be expressed as generalized joins based on an equivalence relation.

Explicit adjacency and Laplacian spectra are derived for these graphs.

Abstract

The zero-divisor graph $\Gamma(R)$ of a ring $R$ is a graph with nonzero zero-divisors of $R$ as vertices and distinct vertices $x,y$ are adjacent if $xy=0$ or $yx=0$. We provide an equivalence relation on a ring $R$ and express $\Gamma(R)$ as a generalized join of graphs on equivalence classes of this relation. We determined the adjacency and Lapalcian spectra of $\Gamma(R)$ when $R$ is a finite semisimple ring.