Computing Wiener and Hyper-Wiener Indices of Zero-Divisor Graph of ℤ ℊ 3 × ℤ I 1 I 2

Abstract

Let $S={\mathbb{Z}}_{{ℊ}^3}\times {\mathbb{Z}}_{{\mathfrak{I}}_1{\mathfrak{I}}_2}$ be a commutative ring where $ℊ,{\mathfrak{I}}_1\mathit{\text{and}}{\mathfrak{I}}_2$ are positive prime integers with ${\mathfrak{I}}_1\ne {\mathfrak{I}}_2$ . The zero-divisor graph assigned to S is an undirected graph, denoted as $Y\left(S\right)$ with vertex set V( $Y$ (S)) consisting of all Zero-divisor of the ring S and for any c, d $\in$ V( $Y$ (S)), $cd\in \text{E}\left(Y\left(\text{S}\right)\right)$ if and only if cd =0. A topological index/descriptor is described as a topological-invariant quantity that transforms a molecular graph into a mathematical real number. In this paper, we have computed distance-based polynomials of $Y\left(\text{R}\right)$ i-e Hosoya polynomial, Harary polynomial, and the topological indices related to these polynomials namely Wiener index, and Hyper-Wiener index.