On The Maximal Wiener Polarity Index Of Trees

The Wiener polarity index Wp(G) of a graph G= (V, E) is the number of unordered pairs of vertices{u, v} of G such that the distance dG(u,v) between u and v is 3. In this paper, the maximal Wiener polarity indices of all trees or chemical trees on some situations are given:(1) If T is a tree with order n and diameter d(d> 5), then with equality if and only if T is a capillary tree CTn(0,…,0,xi,xi+1, xi+2,0,…,0) and 1≤i≤d-5, xi+xi+1+xi+2=n-d-1, xi≥0, xi+2> 0, or(2) Among all trees with n vertices and k pendants, the maximum Wiener polarity index is(3) Among all chemical trees with n> 7 vertices and k> 2 pendants, the maximum Wiener polarity index is...