The characterization of trees with the smaller Wiener polarity indices

Abstract

The Wiener polarity index W _P(G) of a graph G = (V, E) is the number of unordered pairs of vertices {u, v} of G such that the distance d _G(u, v) between u and v is 3. In this paper, we characterize the trees with the second and third smallest Wiener polarity indices among all trees of order n and diameter k.