On the Total Number of Matchings of Trees with Prescribed Diameter

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, the trees with minimal Hosoya index are uniquely determined among all trees in Tn,d.