SPECTRUM OF THE GENERALIZED ZERO-DIVISOR GRAPHS

The generalized zero-divisor graph of a ring $R$, denoted by $\Gamma'(R)$, is a simple (undirected) graph with a vertex set consisting of all nonzero zero-divisors in $R$, and two distinct vertices $x$ and $y$ are adjacent if $x^ny=0$ or $y^nx=0$, for some positive integer $n$. If $R=\displaystyle \prod_{i=1}^{k}R_i$ is a direct product of finite commutative local rings $R_i$ with $|R_i|=p_i^{\alpha_i}$, then we express $\Gamma'(R)$ as a $H$-generalized join of a family $\mathcal F$ of a complete graph and null graphs, where $H$ is a graph obtained from $\Gamma'(S^k)$ by contraction of edges of all nonzero nilpotents at a single vertex ${\bf 0}$, and $S=\{0,1,2\}$ is a multiplicative submonoid of a ring $\mathbb{Z}_4$. Also, we prove that the adjacency spectrum of $\Gamma'(R)$ is $\left\{(-1)^{(\beta-1)}, 0^{(\gamma-3^k+2^k+1)}\right\}\cup\sigma(NA(H))$, where $\beta$ is the number of nonzero nilpotent elements, $\gamma$ is the number of non-nilpotent zero-divisors in $R$ and $N$ is a diagonal matrix whose rows (columns) are indexed with vertices $e\in \Gamma'(H)$ with $e^{th}$ diagonal entry is the cardinality of $e^{th}$ graph in the family $\mathcal F$.