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 P 2 , P 3 , and C 4 . 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. [ABSTRACT FROM AUTHOR]