Hosoya Polynomials of Twisted Toroidal Polyhexes

Abstract

The Hosoya polynomial of a graph \(G\) with vertex set \(V(G)\) is defined as \(H(G, z) = \sum_{u,v \in V(G)} x^{d_G(u,v)}\), where \(d_G(u,v)\) is the distance between vertices \(u\) and \(v\). A toroidal polyhex \(H(p,q,t)\) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, described by a string \((p,q,t)\) of three integers \((p \geq 2, q \geq 1, 0 \leq t \leq p-1)\). In this paper, we derive an analytical formula for calculating the Hosoya polynomial of \(H(p,q,t)\) for \(t = 0\) or \(p\leq 2q\) or \(p \leq q+t\). Notably, some earlier results in [2, 6, 26] are direct corollaries of our main findings.