On the generalized Wiener polarity index of trees with a given diameter

The generalized Wiener polarity index of a graph G, denoted by Wk(G), is the number of unordered pairs of vertices that are at distance k in G. In this paper, we characterize the extremal trees with respect to the index among all trees of order n and diameter d, which partially answers a question of Bollobás and Tyomkyn (2012) and also generalizes some results of Deng et al. (2010).