The Wiener Number in the Context of Generalized Topological Indices

Abstract

A generalized graph-theoretical matrix is used together with two new vectors and a vector-matrix-vector multiplication procedure to generalize the Wiener index of molecular graphs. The distance and adjacency matrices of a graph are particular cases of this novel matrix, which can also be transformed into the inverse distance or Harary matrix. The original Wiener index is represented in an eight dimensional space of the generalized topological indices in the form of a radial graphic. The differences and similarities between the Wiener index and several topological indices are analyzed by using such radial graphics. We studied the following indices: the Zagreb group indices M1 and M2, the Randic χ index, the Balaban J index, and the Harary indices H1 and H2. The generalized approach to topological indices is then applied to optimize the Wiener index to allow for a better description of the boiling points of octane isomers. A great improvement is obtained with a new Wiener index for describing this property and a correlation coefficient of 0.9464 (instead of 0.5392 obtained with the original Wiener index). The results are extended to estimate the boiling points of Cl-Cg alkanes yielding the best model reported up to now with one topological index for this set of compounds. The novel Wiener indices and the original ones are then interpreted in terms of the structural features they account for.