On Wiener and multiplicative Wiener indices of graphs

Let GG be a connected graph of order nn with mm edges and diameter dd. The Wiener index W(G)W(G) and the multiplicative Wiener index π(G)π(G) of the graph GG are equal, respectively, to the sum and product of the distances between all pairs of vertices of GG. We obtain a lower bound for the difference π(G)−W(G)π(G)−W(G) of bipartite graphs. From it, we prove that π(G)>W(G)π(G)>W(G) holds for all connected bipartite graphs, except P2P2, P3P3, and C4C4. We also establish sufficient conditions for the validity of π(G)>W(G)π(G)>W(G) in the general case. Finally, a relation between W(G)W(G), π(G)π(G), nn, mm, and dd is obtained.