Spectral Properties of Zero-Divisor Graphs of Truncated Polynomial Rings

TL;DR

This paper analyzes the spectral properties of zero-divisor graphs derived from truncated polynomial rings, providing explicit spectra and eigenvalue integrality results.

Contribution

It determines the spectra of various matrices associated with zero-divisor graphs of truncated polynomial rings, including adjacency, signless Laplacian, Laplacian, and distance spectra.

Findings

Explicit adjacency spectrum of the zero-divisor graph is obtained.

Signless Laplacian spectrum of the graph is explicitly determined.

Laplacian and distance eigenvalues are all integers.

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} $ \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 $A_{\alpha}$-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.