The maximum Wiener polarity index of trees with pendants

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 work, we give the maximum Wiener polarity index of trees with n vertices and k pendants and find that the maximum value is independent of k when k + 2 ≤ n ≤ 2 k . The corresponding extremal trees are characterized.