On Harary index of graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. For a connected graph G=(V,E) and two nonadjacent vertices v_i and v_j in V(G) of G, recall that G+v_iv_j is the supergraph formed from G by adding an edge between vertices v_i and v_j. Denote the Harary index of G and G+v_iv_j by H(G) and H(G+v_iv_j), respectively. We obtain lower and upper bounds on H(G+v_iv_j)−H(G), and characterize the equality cases in those bounds. Finally, in this paper, we present some lower and upper bounds on the Harary index of graphs with different parameters, such as clique number and chromatic number, and characterize the extremal graphs at which the lower or upper bounds on the Harary index are attained.