On Bicyclic Graphs With Minimal Energies And Hosoya Indices

The HMO totalπ-electron energy is a well-known topological index in theoretical chemistry. In fact, it is in good linear correlation with experimental heats of formation of conjugated hydrocarbons. Furthermore,it can be used to calculate the resonance energies . And the definition can be defined for all graphs. In view of this, we define the energy of a graph, signed E(G), as the sum of the absolute values of all the eigenvalues of graph G. Quite a few lower and upper bounds for E(G) are known[7,8]. On the other hand,very little is known about graphs with extremal energy.Recently the unicyclic graphs and bicyclic graphs with extremal energy become a hot topic. The graphs with extremal energy have been widely discussed, for various classes of unicyclic graphs and bicyclic graphs(see[3,9,10,16,17,26,28,33,36,42,43]).Hosoya topological index, signed z(G), defined as the sum of all the number of k-matchings. Hosoya index, introduced by a Japanese chemist Hosoya, is wildly used molecular descriptions in structure-property-activity studies. The Hosoya index is the first molecular invariant which is found useful in simple regressions. We already have many results(see[2,6,7,10]).Let (?)_n be the class of bicyclic graphs Gonn vertices and containing no disjoint odd cycles of lengths k and l with k +l≡2 (mod 4) Zhang, Zhou[On bicyclic graphs with minimal energies, J. Math. Chem. 37(2005) 423-431] obtained the minimal, secondminimaland third-minimal values on the energies of the graphs in (?)_n and determined the corresponding graphs. In this paper, as the continuance of it, we obtain some new results, as follows. We, firstly, characterized the fourth-minimal, fifth-minimal, sixth-minimal values on the energies and their corresponding graphs. Secondly, we determine the graphs in (?)_n with minimal, second-minimal, third-minimal, fourth-minimal, fifth-minimal, sixthminimal,seventh-minimal, eighth-minimal and ninth-minimal Hosoya indices.