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

Abstract

Let $R$ be a ring with unity. The cozero-divisor graph $\Gamma'(R)$ of a ring $R$ 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 $y \notin Rx$. To extend the corresponding results of the ring $\mathbb{Z}_{n}$ of integer modulo $n$, in this article, 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 classes of rings including the ring $\mathbb{Z}_{n}$ of integers modulo $n$.