Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

Abstract

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} $\Gamma (R)$ is the simple graph with vertex set $V(\Gamma (R))=Z^{\ast }(R)$, 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 $R={\mathbb{Z}}_p[x]/⟨x^c⟩,$ for $c\in \mathbb{N}.$ We determine the spectrum of the $A_{\alpha }$-matrix associated with $\Gamma (R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $\Gamma (R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

arXiv comment38, pages, 2 figures