A class of modified Wiener indices
İvan Gutman, Damir Vukičević, Janez Žerovnik · University of Maribor digital library (University of Maribor) · 2004
The Wiener index of a tree T obeys the relation W(T) = S 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 Nikoli}, Trinajsti} and Randi} put forward a novel modification m W of the Wiener index, defined as m W(T) = S e [ n 1 (e) •n 2 (e) ] -1 .We now extend their definition as m W l (T) = S e [ n 1 (e) •n 2 (e) ] l , and show that some of the main properties of both W and m W are, in fact, properties of m W l , valid for all values of the parameter l ¹0.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 l,Thus m W l 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 l then, in the general case, this ordering is different for different l.