An upper bound for the Harary index of a connected graph and the corresponding extremal graph

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. The graph(s) with maximum Harary index among all graphs with given order and diameter was characterized, and an upper bound for the Harary index in terms of diameter was provided. In addition, the connected graphs of order n with the second maximum and the third maximum Harary indices were characterized, respectively.