Eigenvalues and Wiener index of the Zero Divisor graph $\Gamma[\mathbb {Z}_n]$

TL;DR

This paper investigates the eigenvalues, energy, and spectral properties of zero divisor graphs of certain finite rings, specifically for rings of the form rac{Z}{nZ} with n=p^3 and n=p^2q, providing new insights into their algebraic graph structure.

Contribution

It computes the adjacency matrix, eigenvalues, and energy of zero divisor graphs for specific classes of finite rings, extending spectral graph theory in algebra.

Findings

Eigenvalues of the zero divisor graphs are explicitly determined.

The energy of these graphs is calculated for the given ring classes.

Spectral properties reveal structural insights into the zero divisor graphs.

Abstract

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. In this paper, we consider the zero divisor graph $\Gamma[\mathbb{Z}_n]$ for $n=p^3$ and $n=p^2q$ with $p$ and $q$ primes. We discuss the adjacency matrix and eigenvalues of the zero divisor graph $\Gamma[\mathbb{Z}_n]$. We also calculate the energy of the graph $\Gamma[\mathbb{Z}_n]$.