The extended hyper-Wiener index

According to the definition of molecular connectivity and the definition of a hyper-Wiener index, a novel set of hyper-Wiener indexes (Rn, mRn) are defined and are named the extended hyper-Wiener indexes. Where n = 1, 2, 3, 4,... represents the type of subgraph units and is the number of endmost atoms of the subgraph unit, m is the number of atoms of the subgraph unit. Here n = 1 means the subgraph unit is an atom, n = 2 means the subgraph units are straight-line combinations of m atoms (m = 2, 3, 4, 5, 6,...), and n = 3 means the subgraph units are Y types of combinations of m atoms (m = 4, 5, 6, 7, 8,...), and so on. The potential usefulness of the extended hyper-Wiener index in QSAR and (or) QSPR is evaluated by its correlation with a number of C3–C8 alkanes and by a favorable comparison with models based on the molecular connectivity index and the overall Wiener index. To verify the robustness and the predictive ability of the models, a cross-validation procedure, leave-one-out, and a random test were also performed. The results show that the extended hyper-Wiener indexes examined demonstrate a good potential for QSAR and QSPR studies. Considerably better statistics are obtained when extending the hyper-Wiener index to the extended hyper-Wiener index. The extended hyper-Wiener indexes provided statistical results as good as the molecular connectivity indexes and the overall Wiener index in all models, and the standard deviations provided by these three sets of indexes are rather close. Moreover, this method may provide a better way to apply the Wiener number and the hyper-Wiener index to the system of unsaturated hydrocarbons and organic compounds, including heteroatoms, according to the method of the molecular connectivity index. This can extend the usefulness of the Wiener number and hyper-Wiener index and can make them a kind of widely used topological index in practice.Key words: hyper-Wiener index (R), extended hyper-Wiener index, molecular connectivity index.