Wiener index of the Cozero-divisor graph of a finite commutative ring

TL;DR

This paper derives a formula for the Wiener index of the cozero-divisor graph of finite commutative rings and provides computational tools for specific classes of rings, extending previous results.

Contribution

It introduces a closed-form formula for the Wiener index of the cozero-divisor graph of finite commutative rings and applies it to specific ring classes with computational implementation.

Findings

Derived a closed-form Wiener index formula for finite commutative rings.

Computed Wiener index for rings like $Z_n$ and reduced rings.

Provided SageMath code for Wiener index computation.

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 article, we extend some of the results of [24] to an arbitrary ring. In this connection, we derive a closed-form formula of the Wiener index of the cozero-divisor graph of a finite commutative ring $R$. As applications, we compute the Wiener index of $\Gamma'(R)$, when either $R$ is the product of ring of integers modulo $n$ or a reduced ring. At the final part of this paper, we provide a SageMath code to compute the Wiener index of the cozero-divisor graph of these class of rings including the ring $\mathbb{Z}_{n}$ of integers modulo $n$.