Indices of differences of path lengths: novel topological descriptors derived from electronic interferences in graphs

Novel topological descriptors, namely indices of differences of path lengths (DPs), are deduced from the physical model of wave interferences. Two electrons, moving through a circuit graph within a diffraction experiment, interfere in a given vertex of the graph. It is demonstrated that the overall sum of the inverse of the squares of the differences of topological distances between all pairs of vertices of the graph is a measure of the mean global kinetic energy of the electrons which are able to produce a constructive interference. New topological indices, namely indices of differences of path lengths are thus introduced as derived from such a diffraction pattern. These indices, according to the above expressed, should be a measure of the electron mobility within the molecule. As a consequence, a good prediction is to be expected for properties related to such mobility, such as resonance energy in aromatic hydrocarbons. Our results confirm that in fact, the resonance energies are well predicted by this means. Moreover, the new indices demonstrate to be very useful in the evaluation of biological properties such as antibacterial activities of a wide set of heterogeneous compounds.