Eccentricity Spectra and Integral Eigenvalues of Zero Divisor Graphs

TL;DR

This paper investigates the eccentricity spectra of zero divisor graphs related to rings, revealing integral eigenvalues for certain classes and analyzing their structural properties and energy gaps using matrix analysis.

Contribution

It provides the first detailed analysis of eccentricity spectra of zero divisor graphs, including integral eigenvalues and their relation to graph structure.

Findings

Integral eccentricity eigenvalues for ZDGs of rac{p^t}{p^t} with prime p

Eccentricity spectra computed for specific ZDG classes

Eccentricity energy gaps calculated for applications

Abstract

In this work, we study the eccentricity spectra of zero divisor graphs (ZDGs) associated with the ring $\mathbb{Z}_n.$ While previous studies have examined the Laplacian and distance Laplacian spectra of ZDGs, the eccentricity spectra have remained largely unknown due to the unique features of the eccentricity matrix. More specifically, we prove that for a prime $p$, the ZDG and extended ZDG of $\mathbb{Z}_{p^t}$ have integral eccentricity eigenvalues for $t \geq 3$ and $t \geq 2$, respectively. We also find the eccentricity spectra for specific classes of ZDGs and the relationship between the eccentricity matrix of these ZDGs and their tree structures using matrix analysis tools. In addition, for the usefulness of the energy gap in applications, we have calculated the eccentricity energy gap of ZDGs. These findings reveal interesting behaviours of the eccentricity matrix and may contribute to a more profound understanding of the structural properties of ZDGs.