On Wiener and multiplicative Wiener indices of graphs

Let G be a connected graph of order n with m edges and diameter d. The Wiener index W(G) and the multiplicative Wiener index pi(G) of the graph G are equal, respectively, to the sum and product of the distances between all pairs of vertices of G. We obtain a lower bound for the difference pi(G) - W(G) of bipartite graphs. From it, we prove that pi(G) > W (G) holds for all connected bipartite graphs, except P-2, P-3, and C-4. We also establish sufficient conditions for the validity of pi(G) > W(G) in the general case. Finally, a relation between W(G), pi(G), n, m, and d is obtained. (C) 2016 Elsevier B.V. All rights reserved.