The Harary index of trees
  The Harary index of a graph $G$ is recently introduced topological index, defined on the reverse distance matrix as $H(G)=\sum_{u,v \in V(G)}\frac{1}{d(u,v)}$, where $d(u,v)$ is the length of the shortest path between two distinct vertices $u$ and $v$. We present the partial ordering of starlike trees based on the Harary index and we describe the trees with the second maximal and the second minimal Harary index. In this paper, we investigate the Harary index of trees with $k$ pendent vertices and determine the extremal trees with maximal Harary index. We also characterize the extremal trees with maximal Harary index with respect to the number of vertices of degree two, matching number, independence number, radius and diameter. In addition, we characterize the extremal trees with minimal Harary index and given maximum degree. 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. 
Aleksandar Ili\' c, Guihai Yu and Lihua Feng
math.CO