The maximum Wiener polarity index of trees with kk pendants

The Wiener polarity index Wp(G)Wp(G) of a graph G=(V,E)G=(V,E) is the number of unordered pairs of vertices {u,v}{u,v} of GG such that the distance dG(u,v)dG(u,v) between uu and vv is 3. In this work, we give the maximum Wiener polarity index of trees with nn vertices and kk pendants and find that the maximum value is independent of kk when k+2≤n≤2kk+2≤n≤2k. The corresponding extremal trees are characterized.