Coulson-type integral formulae for the energy changes of graph perturbations

Abstract

The energy of a graph G on n vertices is the sum of the absolute values of the eigenvalues of G. For many molecular graphs, this sum is equivalent to an integral formula used by Coulson to obtain the total π-electron energy of a molecule. Using the resolvent of the adjacency matrix of G, we present integral formulae, of a form similar to that of Coulson, that determine the difference between the energy of G and that of some slight modification applied to G. The graph perturbations considered include the addition of a vertex or edge to G, as well as the removal of a vertex or edge from G. We also provide a decomposition of the energy of G into the sum of n values, where each value is associated with a vertex of G.