Hosoya Polynomials of Circumcoronene Series

Abstract

The Hosoya polynomial (also called Wiener polynomial) of a graph G with the vertex set V(G) is defined as H(G, x) = Sigma({u,v}subset of V(G)) x(dG(u,v)) on variable x, where the sum is over all unordered pairs {u, v} of distinct vertices in G, dG(u,v) is the distance of two vertices. u, v in G. In 2004, Yang and Yeh evaluated H(G,x) for certain graphs of chemical interest, and posted an open problem for evaluating the Hosoya polynomial of the circumcoronene C-r of order r. In this paper, we solve this problem and give analytical expressions associated with r. The topological indices-Wiener index and hyper-Wiener index of C-r can also be obtained from H (C-r, x) through this work.