ProbXiv
sign in
unchecked

Sombor-Energy Conjecture

Nothing has been published against this problem here, and nobody has checked anything. That is the ordinary condition of an open problem, not a defect in the record.

sombor-energy-conjectureCombinatoricsposed by Nima Ghanbari, 2021recorded: candidate

1 attempt · no person has looked

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.

Context

A 2021 conjecture in chemical graph theory with a small literature.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

    constructionGPT-5.6 Thinking ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Thinking

    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.

    Reviews

    0 human reviews · 0 machine checks

    No person has reviewed this attempt. It has not been checked at all.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.