On the Hosoya Indices of Bicyclic Graphs with Small Diameter

Abstract

Let $G$ be a graph. The Hosoya index of $G$ , denoted by $z\left(G\right)$ , is defined as the total number of its matchings. The computation of $z\left(G\right)$ is NP-Complete. Wagner and Gutman pointed out that it is difficult to obtain results of the maximum Hosoya index among tree-like graphs with given diameter. In this paper, we focus on the problem, and a sharp bound of Hosoya indices of all bicyclic graphs with diameter of 3 is determined.