On the Merrifield-Simmons Indices and Hosoya indices of Trees with a Prescribed Diameter

摘要

The Merrifield-Simmons index σ = σ(G) and the Hosoya index z = z(G) of a (molecular) graph G are defined as the total number of the independent vertexsets and the total number of the independent edgesets of the graph G, respectively. Let Fn,d denote the set of trees on n vertices and diameter d. Li, Zhao and Gutman [MATCH Commun. Math. Comput. Chem. 54(2005) 389-402] have determined the uniqiie tree in Fn,d with maximal cr-value. Pan, Xu, Yang and Zhou [MATCH Commun. Math. Comput. Chem., to appear] have recently determined the unique tree in Fn,d with minimal 2-value. In this paper, the first [d/2]+1 Merrifield-Simmons indices and the last [d/2]+1 Hosoya indices of trees in the set Fn,d (3 ≤ d ≤ n - 4) are characterized.