Spectra of the zero-divisor graph of finite rings

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.