Maximal And Minimal λ-Modified Wiener Index For A Given Tree

For each tree T and each real numberλ,λ≠0, theλ-modified Wiener index is definedas ^mW_λ(T)=sum form e∈E(T) [n_T,1(e)·n_T,2(e)]^λ, where n_T,1(e) and n_T,2(e) denote the number ofvertices of T lying on the two sides of the edge e. Let T_n,p be the class of trees with nvertices, p of which are pendent vertices. Let T_n,d be the class of trees with n verticesand diameter d. In this paper, for eachλ≠0 and eachp with 3≤p≤n-2 andeach d with 3≤d≤n-2, we determine the trees in T_n,p with maximal and minimalλ-modified Wiener indices and we determine the trees in T_n,d with minimalλ-modifiedWiener index.