On the Wiener polarity index of trees with maximum degree or given number of leaves

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)=3dG(u,v)=3. In this paper, the minimum (resp. maximum) Wiener polarity index of trees with nn vertices and maximum degree ΔΔ are given, and the corresponding extremal trees are determined, where 2≤Δ≤n−12≤Δ≤n−1. Moreover, the trees minimizing WP(T)WP(T) among all trees TT of order nn and kk leaves are characterized, where 2≤k≤n−12≤k≤n−1.