Eigenvalues of zero-divisor graphs of finite commutative rings

TL;DR

This paper explores the eigenvalues of zero-divisor graphs of finite commutative rings, providing formulas for nullity, spectra of specific rings, and a graph product that relates ring structure to graph eigenvalues.

Contribution

It introduces a new graph product and derives explicit eigenvalue formulas, linking ring properties with graph spectra in finite commutative rings.

Findings

Formulas for the nullity of zero-divisor graphs.

Explicit spectra for specific product rings.

A new graph product relating ring structure to eigenvalues.

Abstract

We investigate eigenvalues of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $\Gamma(R)$. The graph $\Gamma(R)$ is defined as the graph with vertex set consisting of all non-zero zero-divisors of $R$ and adjacent vertices $x,y$ whenever $xy = 0$. We provide formulas for the nullity of $\Gamma(R)$, i.e. the multiplicity of the eigenvalue 0 of $\Gamma(R)$. Moreover, we precisely determine the spectra of $\Gamma(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ and $\Gamma(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ for a prime number $p$. We introduce a graph product $\times_{\Gamma}$ with the property that $\Gamma(R) \cong \Gamma(R_1) \times_{\Gamma} \ldots \times_{\Gamma} \Gamma(R_r)$ whenever $R \cong R_1 \times \ldots \times R_r.$ With this product, we find relations between the number of vertices of the zero-divisor graph $\Gamma(R)$, the compressed zero-divisor graph, the structure of the ring $R$ and the eigenvalues of $\Gamma(R)$.