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 ( Z n ) for n = p N 1 q N 2 , where p < q are primes and N 1 , N 2 are positive integers.