A class of modified Wiener indices

The Wiener index of a tree T obeys the relation W(T) = Σ e n 1 (e) . n 2 (e) where n 1 (e) and n 2 (e) are the number of vertices on the two sides of the edge e, and where the summation goes over all edges of T. Recently Nikolic, Trinajstic and Randic put forward a novel modification m W of the Wiener index, defined as m W(T) = Σ e [n 1 (e) .n 2 (e)] - 1 . We now extend their definition as m W λ (T) = Σ e [n 1 (e) .n 2 (e)] λ , and show that some of the main properties of both W and m W are, in fact, properties of m W λ , valid for all values of the parameter λ≠0. In particular, if T n is any n-vertex tree, different from the n-vertex path P n and the n-vertex star S n , then for any positive λ, m W λ (P n ) > m W λ (T n ) > m W λ (S n ), whereas for any negative λ, m W λ (P n ) < m W λ (T n ) < m W λ (S n ). Thus m W λ provides a novel class of structure-descriptors, suitable for modeling branching-dependent properties of organic compounds, applicable in QSPR and QSAR studies. We also demonstrate that if trees are ordered with regard to m W λ then, in the general case, this ordering is different for different λ.