On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Given a commutative ring R with identity 1≠0, let the set Z (R) denote the set of zero-divisors and let Z* (R) =Z (R) ∖0 be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ (R), is a simple graph whose vertex set is Z* (R) and each pair of vertices in Z* (R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ (Zn) for n=pN1qN2, where p<q are primes and N1, N2 are positive integers.