Minimizing a class of unicyclic graphs by means of Hosoya index

The Hosoya index, denoted by z ( G ) , of a (molecular) graph G is defined as the total number of independent-edge sets of G . Let U n be the set of unicyclic graphs with n vertices. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than 3 in its unique cycle. Denote by U n 1 the set of fully loaded unicyclic graphs. In this paper, graphs in U n 1 with minimal, second-minimal and third-minimal Hosoya indices are uniquely determined, respectively.