On the eigenvalues of zero-divisor graph associated to finite commutative ring

Abstract

Let Z(R) be the set of zero-divisors of a commutative ring R with non-zero identity and (Formula presented.) be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by (Formula presented.) is a simple graph whose vertex set is (Formula presented.) and two vertices (Formula presented.) are adjacent if and only if (Formula presented.) In this paper, we investigate the adjacency matrix and the spectrum of the zero-divisor graphs (Formula presented.) for (Formula presented.) where (Formula presented.) are primes and M, N are positive integers. Moreover, we obtain the clique number, stability number and girth of (Formula presented.).