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Sombor-Energy Conjecture

Combinatorics · posed by Nima Ghanbari, 2021 · candidate

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Statement

Does every nontrivial finite simple graph have noninteger Sombor energy? If ρ1,…,ρn\rho_1,\ldots,\rho_n are the eigenvalues of the Sombor matrix of a graph GG, its Sombor energy is

ESO(G)=∑i=1n∣ρi∣.E_{\mathrm{SO}}(G)=\sum_{i=1}^{n}|\rho_i|.

The conjecture asserted that ESO(G)∉ZE_{\mathrm{SO}}(G)\notin\mathbb Z for every nontrivial graph. A connected graph on nine vertices is exhibited with ESO(G)=64E_{\mathrm{SO}}(G)=64, disproving the conjecture.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    GPT-5.6 Thinking

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    The AI constructed a connected bipartite graph on nine vertices and calculated its Sombor spectrum exactly. Writing its Sombor matrix in the block form

    S(G)=(0BBT0),S(G)=\begin{pmatrix}0&B\\B^{T}&0\end{pmatrix},

    the singular values of BB were found to be

    5,5,11+31,11−31.5,\quad 5,\quad 11+\sqrt{31},\quad 11-\sqrt{31}.

    Therefore,

    ESO(G)=2(5+5+(11+31)+(11−31))=64.E_{\mathrm{SO}}(G) =2\left(5+5+(11+\sqrt{31})+(11-\sqrt{31})\right) =64.

    The AI also audited the edge list, degrees, connectivity, bipartition, matrix multiplication, characteristic polynomial, singular values and final energy calculation, and produced a self-contained proof.

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