Assessment of Distance Related Spectrum of the Zero-divisor Graph

Abstract

This paper aims to find the distance, distance Laplacian and distance signless Laplacian of \(\Gamma\left(Z_{n}\right)\),for some values of n. For a commutative ring  with non-zero identity, \(Z^{*}(R)\) denote the set of nonzero zero-divisors of R . The zero-divisor graph of R denoted by \(\Gamma\) (R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices \(x, y \in Z^{*}(R)\) are adjacent if and only if xy = 0 . In this paper, the distance, distance Laplacian and the distance singless Laplacian spectrum of \(\Gamma\left(Z_{n}\right)\) for n = p³ , pq are investigated. The specific combinatorial structure as well as the typical block structure of the distance related matrices of the graphs mentioned in this study both contributed to our decision to use matrices in the spectrum computation.