On the Spectrum of Generalized Zero-Divisor Graph of the Ring \(\mathbb{Z}_{p^\alpha q^\beta}\)

The generalized zero-divisor graph of a commutative ring \(R\), denoted by \(\Gamma'(R)\), is a simple (undirected) graph with vertex set \(Z^*(R)\), the set of all nonzero zero-divisors of \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if \(x^ny=0\) or \(y^nx=0\), for some positive integer \(n\). In this paper, we determine the adjacency spectrum of \(\Gamma'(\mathbb{Z}_{p^{\alpha}q^{\beta}})\), where \(p,q\) are distinct primes and \(\alpha, \beta\) are positive integers. Also, we obtain the clique number, stability number, diameter, and the girth of \(\Gamma'(\mathbb{Z}_{p^{\alpha}q^{\beta}})\).