Hosoya index of thorny polymers

Abstract

A matching in a graph G is a collection of edges of G such that no two of them share a vertex. The number of all matchings in G is called its Hosoya index. In this paper, we compute Hosoya indices of several classes of unbranched polymers made of cycles of the same lengths arranged around a middle path and decorated by attaching to each vertex, a given number of pendent vertices or thorns. We establish linear recurrences satisfied by those numbers and obtain explicit formulas in terms of Fibonacci polynomials and their generalizations. Some possible directions of future research are also indicated.