Energies of Graphs and Molecules

Abstract

This contribution highlights the two different definitions of ‘energy’ that are used in Huckel theory and graph theory. In Huckel Theory, with appropriate choices of units and scale, the energy of a π system is a partial sum of adjacency eigenvalues, weighted by occupancy, whereas the ‘graph energy’ is the full sum unweighted over adjacency eigenvalues, taking absolute values. The two definitions give different results for many non‐bipartite systems, with the graph energy always larger than or equal to the Huckel energy for all electron counts. Infinite families of graphs where Huckel and graph energies are equal include the fully vertex truncated cubic polyhedra, and the spheriphane decorations of the cubic polyhedra. However, the discrepancy between the two energies can also be made arbitrarily large, by taking for example, graphs from the diamond‐necklace family. Most fullerenes are discrepant graphs.