On the Spectrum of Generalized Zero-Divisor Graph of the Ring Zpαqβ

Abstract

The generalized zero-divisor graph of a commutative ring $R$, denoted by ${\mathrm{\Gamma }}^{\prime }(R)$, is a simple (undirected) graph with vertex set $Z^{\ast }(R)$, the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if $x^{n}y=0$ or $y^{n}x=0$, for some positive integer $n$. In this paper, we determine the adjacency spectrum of ${\mathrm{\Gamma }}^{\prime }({\mathbb{Z}}_{p^{\alpha }{q}^{\beta }})$, 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 ${\mathrm{\Gamma }}^{\prime }({\mathbb{Z}}_{p^{\alpha }{q}^{\beta }})$.