The smallest Hosoya index of unicyclic graphs with given diameter

The Hosoya index of a (molecular) graph is defined as the total number of the matchings, including the empty edge set, of this graph. Let ${cal{U}}_{n,d}$ be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from ${cal{U}}_{n,d}$ minimizing the Hosoya index and determine the values of corresponding indices. Moreover, the third smallest Hosoya index of unicyclic graphs is determined