Wiener index of local rings

Let $R$ be a finite commutative ring with nonzero identity. Let $Z^{*}(R)$ be the set of nonzero zero-divisors of $R$. We are dealing with the zero-divisor graph of $R$ which is denoted by $\Gamma(R)$ with vertex set $Z^{*}(R)$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. The motivation of this study is to compute Wiener index in algebraic graph theory for special type of graph called zero-divisor graph. Wiener index is defined as the sum of all distance between all pairs of vertices in $\Gamma(R)$. In addition, we generalize the Wiener index of the zero-divisor graph in $\mathbb{Z}_p[x]/(x^2)$ for any prime number $p$. We obtain our results and methods by tables and figures.