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 π(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 π(G)-W(G) of bipartite graphs. From it, we prove that π(G)>W(G) holds for all connected bipartite graphs, except _P2, _P3, and _C4. We also establish sufficient conditions for the validity of π(G)>W(G) in the general case. Finally, a relation between W(G), π(G), n, m, and d is obtained.