The Harary index of digraphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph and named in honor of Professor Frank Harary. For a connected graph G = (V, E) with edge connectivity lambda(G) >= 2, and an edge v(i)v(j) epsilon E(G), G - v(i)v(j) is the subgraph formed from G by deleting the edge vivj. Denote the Harary index of G and G - v(i)v(j) by H(G) and H(G -v(i)v(j)). Xu and Das [K.X. Xu, K.C. Das, On Harary index of graphs, Dicrete Appl.Math. 159 (2011) 1631-16401 obtained lower and upper bounds on H(G + v(i)v(j)) - H(G) and characterize the equality cases in those bounds. We find that the equality case in lower bound is not true and we correct it. In this paper, we give lower and upper bounds on H(G) - H(G - v(i)v(j)), and give some graphs to satisfy the equality cases in these bounds. Furthermore, we extend the Harary index to the directed graphs and get similar conclusions.