Minimal Hosoya Index Figure Unicyclic Graphs On Full Suspension

In molecular structure analysis,a molecular topological structure may be modelledas an undirected graph. A topological index is a map from the set of chemical com-pounds represented by molecular graphs to the set of real number. Experimental resultsshowed that many topological indices are closely correlated with some physicochemicalcharacteristics.The Hosoya index is one of the popular and valuable topological indexin chemistry molecular graphs thoery.Let G be a graph with n vertices.Its Hosoya index Z(G) ,is defined to be the totalnumber of the matchings of G ,namely, Z(G) =∑s=0[n/2] m(G, s) ,where m(G, s) is thenumber of s-matchings of G.An s-matching of a graph G is a subset M of its edgeset with the property that |M| = s and M contains no two edges sharing a commonvertex.For convenience and consistence, it will be always assumed that m(G, 0) = 1.Let un be the set of unicyclic graphs with n vertices.A fully loaded unicyclic graphis a unicyclic graph with the property that there exists no vertex with degree less than 3in its unique cycle.Let un1 be the set of fully loaded unicyclic graphs.Let un(l) and u1n(l)denote resp.the subset of un(l) and u1n(l) in which every graph has a unique cycle oflength l.In this paper, o ur main aim is to investigate the Hosoya of unicyclic graphs.Weobtain the following results.(1)The fourth-minimal value of Hosoya of graphs in un1 are C_n~3(1, 4, n - 8)(12≤n≤14) and (G|~)(n≥15).(2)The fifth-minimal value of Hosoya of graphs in un1 is C_n~3(2, 2, n - 7).(3)The sixth-minimal value of Hosoya of graphs in un1 is G˙.