On Modified and Reverse Wiener Indices of Trees

The Wiener index is a well-known measure of graph or network structures with similarly useful variants of modified and reverse Wiener indices. The Wiener index of a tree T obeys the relation W(T)= nT,1(e)·nT,2(e) where nT,1(e) and nT,2(e) are the number of vertices of T lying on the two sides of the edge e, and where the summation goes over all edges of T. The λ -modified Wiener index is defined as mWλ (T) = [nT,1(e)·nT,2(e)]λ . For each λ > 0 and each integer d with 3 ≤ d ≤ n− 2, we determine the trees with minimal λ -modified Wiener indices in the class of trees with n vertices and diameter d. The reverse Wiener index of a tree T with n vertices is defined as Λ(T)=½n(n-1)d(T)-W(T), where d(T) is the diameter of T. We prove that the reverse Wiener index satisfies the basic requirement for being a branching index.