The largest eigenvalues of adjacency and Laplacian matrices, and ionization potentials of alkanes

Abstract

If <i>G </i>is a molecular graph and <i>A</i>(<i>G</i>) its adjacency matrix, then the Laplacian matrix is defined as <i>L</i>(<i>G</i>)<i>=D</i>(<i>G</i>)<i>-A</i>(<i>G</i>)<i> </i>where <i>D</i>(<i>G</i>)<i> </i>is the diagonal matrix of vertex degrees. We establish a relation between the largest eigenvalues of <i>A(G)</i> and<i> L</i>(<i>G</i>)<i> </i>in case of the molecular graphs of alkanes. This relation is linear, but differs for alkanes with and without a quaternary<i> </i>carbon atom, revealing the main features of the structure-dependence of their (first) ionization potential. An analogous, yet<i> </i>more general , relation holds for all trees.