Overall connectivity--a next generation molecular connectivity

The development of molecular connectivity concept and some of its key elements - Randić's inverse-square-root function and the detailed subgraph characterization - are analyzed. The concept of overall connectivity recently advanced is presented as a next step in unfolding the ideas of molecular connectivity by combining them with those of molecular complexity. Definitions of overall connectivity index, eth-order overall connectivities, and overall connectivity vector are presented along with formulae for calculating these sets of topological indices for several classes of graphs of chemical relevance. Based on sums of adjacencies over all subgraphs (or up to a limiting subgraph size in large molecules), the overall connectivities increase both with molecule size and complexity, as expressed in branching and cyclicity of molecular skeleton. When applied to molecules containing heteroatoms, valence overall connectivities are constructed employing the Kier and Hall scheme. The usefulness of the novel indices is demonstrated by modeling physicochemical properties of alkane compounds. A detailed comparison is made with other models derived for the same set of compounds, proceeding from molecular connectivity, as well as with two other probe connectivity functions--the overall connectivity versions of the second Zagreb index, and a derivative inverse function of this index. The favorable comparisons indicate the need of molecular connectivity paradigm revisiting, and show the potential of the overall connectivity indices for QSPR/QSAR applications.