Sombor energy and Hückel rule

Abstract

The Sombor index, a recently invented vertex-degree-based graph invariant, is insensitive to the size of cycles contained in a graph. In contrast to this, the Sombor energy, the sum of absolute values of the Sombor matrix, is found to have a significant cycle-size dependence. In the case of bipartite graphs, this dependence is analogous to the Hückel (4n + 2)-rule: cycles of size 4, 8, 12, … decrease, and cycles of size 6, 10, 12, … increase the Sombor energy. A theorem corroborating this empirical observation is offered.