A Note on Wiener and Hyper-Wiener Indices of Abid-Waheed Graph

A Hosoya polynomial is a polynomial connected to a molecular graph, which is a graph representation of a chemical compound with atoms as vertices and chemical bonds as edges. A graph invariant is the Hosoya polynomial; it is a graph attribute that does not change under graph isomorphism. It provides information about the number of unique non-empty subgraphs in a given graph. A molecular graph's size and branching complexity are determined by a topological metric known as the Wiener index. The Wiener index of each pair of vertices in a molecular network is the sum of those distances. The topological index, one of the various classes of graph invariants, is a real number related to a connected graph's structure .The goal of this article is to compute the Hosoya polynomial of some class of Abid-Waheed graph. Further, this research focused on a C++ algorithm to calculate the wiener index of AW(m)(9) and AW(m)(11). The Wiener index ((WI)-I-& lowast;) and Hyper-Wiener index ((HWI)-W-& lowast;-I-& lowast;) are calculated using Hosoya polynomial (H-& lowast;-polynomial) of some family of Abid-Waheed graphs AW(m )(9)and AW(m)(11). Illustrations and applications are given to enhance the research work.