Hosoya polynomial of zigzag polyhex nanotorus

Abstract

The Hosoya polynomial of a molecular graph G is defined as H(G,λ) = ∑{u, v}⊆V(G)λd(u,v), where d(u,v) is the distance between vertices u and v. The first derivative of H(G,λ) at λ = 1 is equal to the Wiener index of G, defined as W(G) = ∑ {u, v}⊆V(G)d(u,v). The second derivative of 1/2λH(G, λ) at λ = 1 is equal to the hyper-Wiener index, defined as WW(G) = 1/2W(G)+1/2∑{u, v}⊆V(G)d(u,v)2. Xu et al. 1 computed the Hosoya polynomial of zigzag open-ended nanotubes. Also Xu and Zhang2 computed the Hosoya polynomial of armchair open-ended nanotubes. In this paper, a new method was implemented to find the Hosoya polynomial of G = HC6[p,q], the zigzag polyhex nanotori and to calculate the Wiener and hyper Wiener indices of G using H(G,λ).