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...

let g be a simple graph. the hosoya polynomial of g is ( , ) ,( , ) = { , } ( ) xd u v h g x  u v v gwhere d(u,v) denotes the distance between vertices u and v . the dendrimer nanostar is apart of a new group of macromolecules. in this paper we compute the hosoya polynomial foran infinite family of dendrimer nanostar. as a consequence we obtain the wiener index andthe hyper-wiener index of th...

The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K{_2}×H, where H is a path graph P{_n}, cyclic graph C{_n}, or star gra...

We characterize the trees T with n vertices whose Hosoya index (total number of matchings) is Z(T ) > 16fn−5 resp. the trees whose Merrifield-Simmons index (total number of independent subsets) is σ(T ) < 18fn−5 + 21fn−6, where fk is the kth Fibonacci number. It turns out that all the trees satisfying the inequality are tripodes (trees with exactly three leaves) and the path in both cases. Furt...

Let T be an acyclic graph without perfect matching and Z(T ) be its Hosoya index; let Fn be the nth Fibonacci number. It is proved in this work that Z(T ) ≤ 2F2m F2m+1 when T has order 4m with the equality holding if and only if T = T1,2m−1,2m−1, and that Z(T ) ≤ F2 2m+2 + F2m F2m+1 when T has order 4m + 2 with the equality holding if and only if T = T1,2m+1,2m−1, where m is a positive integer ...