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 calUn,d be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from calUn,d minimizing the Hosoya index and determine the values of corresponding indices.
Moreover, the third smallest Hosoya index of unicyclic graphs is determined