The relation between the eigenvalue sum and the topological index Z revisited

The topological index Z (put forward by one of the present authors, Bull. Chem. Soc. Jpn., 44, 2332 (1971)) has proved to be a structure-descriptor suitable for modeling physico-chemical properties of both saturated and conjugated unsaturated organic compounds. A linear relation between log Z and the Huckel molecular orbital total pi-electron energy E-pi was established a long time ago (Theor. Chim. Acta 38, 37 (1975)), but was restricted to pi-electron species without non-bonding molecular orbitals (n(0) = 0, where n(0) is the number of zero eigenvalues of the molecular graph). Recently a novel structure-descriptor ES was put forward, the "eigenvalue sum" equal to E-pi/2, but applicable to saturated hydrocarbons. In this case no need not be zero. We demonstrate that, for each group of alkane isomers with a fixed value of n(0), there is a linear relation between ES and log Z.