  The Harary index of a graph is defined as the sum of reciprocals of distances
between all pairs of vertices of the graph. In this paper we provide an upper
bound of the Harary index in terms of the vertex or edge connectivity of a
graph. We characterize the unique graph with maximum Harary index among all
graphs with given number of cut vertices or vertex connectivity or edge
connectivity. In addition we also characterize the extremal graphs with the
second maximum Harary index among the graphs with given vertex connectivity.
