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 of zero-divisors and Z*(R)=Z(R)?{0} be non-zero R. The zero-divisor graph R, denoted by ?(R), is simple whose vertex Z*(R) each pair vertices in are adjacent when their product 0. In this article, we find structure Laplacian spectrum graphs ?(Zn) for n=pN1qN2, where p<q primes N1,N2 positive integers.