Problems / combinatorics

combinatorics / Spectral graph theory

Sombor-Energy Conjecture

Does every nontrivial finite simple graph have noninteger Sombor energy? If $\rho_1,\ldots,\rho_n$ are the eigenvalues of the Sombor matrix of a graph $G$, its Sombor energy is $$E_{\mathrm{SO}}(G)=\sum_{i=1}^{n}|\rho_i|.$$ The conjecture asserted that $E_{\mathrm{SO}}(G)\notin\mathbb Z$ for every nontrivial graph. A connected graph on nine vertices is exhibited with $E_{\mathrm{SO}}(G)=64$, disproving the conjecture.