On the structure-dependence of the largest eigenvalue of the distance matrix of an alkane

Abstract

The distance matrix of an alkane has a unique positive eigenvalue A., which has recently been applied as a topological index and used for ordering alkane isomers with respect to their branching and for calculating their boiling points. We have now determined the main structural features of the alkane molecule, on which A depends. These are the number of vertices of the molecular graph (n) and the sum of the squares of the distances between all pairs of vertices (<em>S</em>). A good linear correlation between Ʌ and (n<em>S</em>)^1/2 is shown to exist. Lower and upper bounds of Ʌ are deduced, both depending solely on n and S. Because <em>S</em> = 2 WW - W, there exists a connection between A, the Wiener index (W) and the hyper- Wiener index (WW).