Minimizing a class of unicyclic graphs by means of Hosoya index

The Hosoya index, denoted by z(G)z(G), of a (molecular) graph GG is defined as the total number of independent-edge sets of GG. Let UnUn be the set of unicyclic graphs with nn 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 Un1 the set of fully loaded unicyclic graphs. In this paper, graphs in Un1 with minimal, second-minimal and third-minimal Hosoya indices are uniquely determined, respectively.