A note on eigenvalues of zero divisor graphs associated with commutative rings

TL;DR

This paper investigates the eigenvalues and topological indices of zero divisor graphs of certain commutative rings, correcting previous results and establishing new properties related to energy and Zagreb indices.

Contribution

It corrects earlier findings on eigenvalues and Zagreb indices of zero divisor graphs and provides new formulas and properties for these graphs of specific rings.

Findings

Corrected eigenvalues and energy of zero divisor graphs.

Proved non-hyperenergetic and hypoenergetic properties for certain rings.

Derived formulas for topological and Zagreb indices.

Abstract

For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $\Gamma(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(\Gamma(R))$ are adjacent if and only if $x\cdot y=0.$ In this note, we provide counter examples to the eigenvalues, the energy and the second Zagreb index related to zero divisor graphs of rings obtained in [Johnson and Sankar, J. Appl. Math. Comp. (2023), \cite{johnson}]. We correct the eigenvalues (energy) and the Zagreb index result for the zero divisor graphs of ring $\mathbb{Z}_{p}[x]/\langle x^{4} \rangle.$ We show that for any prime $p$, $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is non-hyperenergetic and for prime $p\geq 3$, $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ is hypoenergetic. We give a formulae for the topological indices of $\Gamma(\mathbb{Z}_{p}[x]/\langle x^{4} \rangle)$ and show that its Zagreb indices satisfy Hansen and Vuki$\check{c}$cevi\'c conjecture \cite{hansen}.