On the spectrum of generalized zero-divisor graph of the ring $\mathbb Z_{p^\alpha q^\beta}$

Abstract

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 $\Gamma'(\mathbb Z_{p^{\alpha}q^{\beta}})$.