Graph Theory and Molecular Orbitals. XVIII. On Topological Resonance Energy

Mathematical relations necessary for the calculation of the previously introduced15 topological resonance energy, TRE, are derived. TRE, as defined in ref. 15, was rather inconvenient for numerical calculations. In the present work these computing difficulties are overcome. Acyclic characteristic polynomial pac and acyclic spectrum of a graph are defined and their close relationship to TRE is demonstrated. Hence, the evaluation of pac is the crucial .step in the computation of TRE's. A graphical procedure is described and exemplified for the determination of P"c's both for simple graphs and for graphs with loops and weighted edges.