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

摘要

The zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices are the nonzero zero divisors in R and two vertices x and y are adjacent if and only if xy=0. We study the adjacency and Laplacian eigenvalues of the zero-divisor graph Γ(R) of a finite commutative von Neumann regular ring R. We prove that Γ(R) is a generalized join of its induced subgraphs. Among the |Z(R)∗| eigenvalues (respectively, Laplacian eigenvalues) of Γ(R), exactly |B(R)|−2 are the eigenvalues of a matrix obtained from the adjacency (respectively, Laplacian) matrix of Γ(B(R))-the zero-divisor graph of nontrivial idempotents in R. We also determine the degree of each vertex in Γ(R), hence the number of edges.