Spectrum of the zero-divisor graph of von Neumann regular rings

Abstract

The zero-divisor graph [Formula: see text] of a commutative ring [Formula: see text] is the graph whose vertices are the nonzero zero divisors in [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We study the adjacency and Laplacian eigenvalues of the zero-divisor graph [Formula: see text] of a finite commutative von Neumann regular ring [Formula: see text]. We prove that [Formula: see text] is a generalized join of its induced subgraphs. Among the [Formula: see text] eigenvalues (respectively, Laplacian eigenvalues) of [Formula: see text], exactly [Formula: see text] are the eigenvalues of a matrix obtained from the adjacency (respectively, Laplacian) matrix of [Formula: see text]-the zero-divisor graph of nontrivial idempotents in [Formula: see text]. We also determine the degree of each vertex in [Formula: see text], hence the number of edges.