Ordering trees having small reverse wiener indices

Abstract

The Wiener index W(G) of a connected graph G is defined as the sum of distances between all unordered pairs of vertices of G. As a variation of the Wiener index, the reverse Wiener index of G is defined as ?(G) = ? n(n ? 1)d ? W(G), where n is the number of vertices, and d is the diameter of G. It is known that the star is the unique n-vertex tree with the smallest reverse Wiener index. We now determine the second and the third smallest reverse Wiener indices of n-vertex trees, and characterize the trees whose reverse Wiener indices attain these values for n ? 5.