Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two distinct vertices$x,y\in Z^{\ast}(R)$ are adjacent if and only if $xy=0$ in $R$. In this paper we investigate the zero-divisor graph of the truncated polynomial ring $R=\mathbb{Z}_{p}[x]/\langle x^{c}\rangle,$ for $c\in\mathbb{N}.$ We determine the spectrum of the $A_α$-matrix associated with $ \varGamma(R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $ \varGamma(R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

翻译：设$R$为含单位元的交换环，$Z^{\ast}(R)$表示$R$中所有非零零因子的集合。\emph{零因子图}$\varGamma(R)$是以$V(\varGamma(R))=Z^{\ast}(R)$为顶点集的简单图，其中两个不同顶点$x,y\in Z^{\ast}(R)$相邻当且仅当$xy=0$在$R$中成立。本文研究截断多项式环$R=\mathbb{Z}_{p}[x]/\langle x^{c}\rangle$（其中$c\in\mathbb{N}$）的零因子图。我们确定了与$\varGamma(R)$相关联的$A_α$-矩阵的谱，并作为特例，显式地得到了$\varGamma(R)$的邻接谱和无符号拉普拉斯谱。进一步地，我们证明了这些图的拉普拉斯特征值以及距离特征值均为整数。