The generalized Hosoya triangle is an arrangement of numbers where each entry is a product of two generalized Fibonacci numbers. We define a discrete convolution 1 C based on the entries of the generalized Hosoya triangle. We use C and generating functions to prove that the sum of every k-th entry in the n-th row or diagonal of generalized Hosoya triangle, beginning on the left with the first e...

We study the number of independent vertex subsets (known as the MerrifieldSimmons index in mathematical chemistry) and the number of independent edge subsets (called the Hosoya index) for trees whose vertex degrees are restricted to 1 or d (for some d ≥ 3), a natural restriction in the chemical context. We find that the minimum of the Merrifield-Simmons index and the maximum of the Hosoya index...

The Merrifield-Simmons index i G of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets of G The Hosoya index z G of a graph G is defined as the total number of independent edge subsets, that is, the total number of its matchings. By C n, k, λ we denote the set of graphs with n vertices, k cyc...

Abstract The n-th order Wiener index of a molecular graph G was put forward by Estrada et al. [New J. Chem. 22 (1998) 819] as ( ) 1 ( , ) n n x W H G x where ( , ) H G x is the Hosoya polynomial. Recently Brückler et al. [Chem. Phys. Lett. 503 (2011) 336] considered a related graph invariant, ( ) 1 1 (1/ !) ( ( , )) / n n n n x W n d x H G x d x . For n=1, both W and W reduce to the ordinary W...

Akio Hosoya, Thomas Buchert, 1, 3 and Masaaki Morita 5 Department of Physics, Tokyo Institute of Technology, Oh–Okayama, Meguro–ku, Tokyo 152–0033, Japan Theoretische Physik, Ludwig–Maximilians–Universität, Theresienstr. 37, D–80333 München, Germany Department of Physics and Research Center for the Early Universe (RESCEU), School of Science, The University of Tokyo, Tokyo 113–0033, Japan Depart...

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method...

the hosoya index and the merrifield-simmons index are two types of graph invariants used in mathematical chemistry. in this paper, we give some formulas for computed these indices for some classes of corona product and link of two graphs. furthermore, we obtain exact formulas of hosoya and merrifield-simmons indices for the set of bicyclic graphs, caterpillars and dual star.

Let G = (V (G), E(G)) be a graph. An m−matchings of G is a set of edges of size m in which any two edges are mutually independent. Denote by z(G,m) the number of m−matchings of G. Let z(G) be the total number of matchings in G, namely z(G) = bn 2 c ∑ m=1 z(G,m). It’s well-known that z(G) are also named as Hosoya index. Let Tn,d be the set of trees of on n vertices with diameter d. In this paper...