The Harary index of trees

Abstract

The Harary index of a graph G is recently introduced topological index, defined on the reverse distance matrix as H(G) - Σ _Ju,v∈V(G)1/d(u,v), where d(u,v) is the length of the shortest path between two distinct vertices u and v. In this paper, we investigate the Harary index of trees with given number of pendent vertices, number of vertices of degree two, matching number, independence number, maximum vertex degree, radius and diameter. We concluded that in all presented classes, the trees with maximal Harary index are exactly those trees with the minimal Wiener index, and vice versa.