Upgrading the Wiener index

Abstract

The Wiener index W is the oldest molecular-graph-based structure-descriptor. It is defined as the sum of the distances of all pairs of vertices of the molecular graph G, where the distance is the number of edges in the shortest path connecting the respective vertices, and where G is the hydrogen-depleted molecular graph. This seemingly very simple topological index could be "upgraded" (a) by using as the distance the sum of the bond lengths along the shortest path, or (b) by using the Euclidean distance between the respective pairs of atoms. Each of these "upgraded" Wiener indices could be computed either (α) for the hydrogen-depleted or (β) for the hydrogen-filled molecular graph. We provide examples showing that none of the modifications (aα), (aβ), (bα), (bβ) yields better results than the ordinary Wiener index, and that there is a very good linear correlation between W and its "upgraded" variants.