Harary and Hosoya Polynomials of Some Chain Networks

Abstract

Chemical graph theory is a branch of mathematics that computes topological indices-molecular descriptors derived from the graph representation of a chemical compound. In 1988, Hosoya introduced the Hosoya polynomial of a graph, which has the distinctive ability to sum the number of paths of varying distances within the graph. To better understand the geometric patterns underlying the computed distance-based topological indices and the Hosoya polynomial of the network, we present graphical illustrations. This analysis of the Hosoya polynomial and its associated indices lays the groundwork for further investigations into chain hex-derived networks and their properties. In this article, we derive the closed-form expressions for the Wiener, hyper-Wiener, and Harary indices, as well as the general form of the Hosoya polynomial, for three specific network types: the Chain Oxide, Chain Silicate, and Chain Hex-Derived Network of Type 3.