On the cozero-divisor graphs assosciated to rings

TL;DR

This paper investigates the spectral properties, connectivity, and Wiener index of cozero-divisor graphs associated with rings, especially focusing on rings of integers modulo n, revealing new spectral and structural insights.

Contribution

It provides the first detailed spectral analysis of cozero-divisor graphs for rings, characterizes when spectral radius equals graph order, and computes the Wiener index for these graphs.

Findings

Laplacian spectrum of $ ext{Coz}(bZ_n)$ computed for specific n

Identified conditions when spectral radius equals graph order

Derived Wiener index for arbitrary n

Abstract

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$, denoted by $\Gamma'(R)$, is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x \notin Ry$ and $y \notin Rx$. In this paper, first we study the Laplacian spectrum of $\Gamma'(\mathbb{Z}_n)$. We show that the graph $\Gamma'(\mathbb{Z}_{pq})$ is Laplacian integral. Further, we obtain the Laplacian spectrum of $\Gamma'(\mathbb{Z}_n)$ for $n = p^{n_1}q^{n_2}$, where $n_1, n_2 \in \mathbb{N}$ and $p, q$ are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of $\Gamma'(\mathbb{Z}_n)$, we characterized the values of $n$ for which the Laplacian spectral radius is equal to the order of $\Gamma'(\mathbb{Z}_n)$. Moreover, the values of $n$ for which the algebraic connectivity and vertex connectivity of $\Gamma'(\mathbb{Z}_n)$ coincide are also described. At the final part of this paper, we obtain the Wiener index of $\Gamma'(\mathbb{Z}_n)$ for arbitrary $n$.