VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
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.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
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
$$S(G)=\begin{pmatrix}0&B\\B^{T}&0\end{pmatrix},$$
the singular values of $B$ were found to be
$$5,\quad 5,\quad 11+\sqrt{31},\quad 11-\sqrt{31}.$$
Therefore,
$$E_{\mathrm{SO}}(G)
=2\left(5+5+(11+\sqrt{31})+(11-\sqrt{31})\right)
=64.$$
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.
The proof has been internally audited using exact calculations. The graph has nine vertices, fifteen distinct edges and degree sequence $(4,4,4,3,3,3,3,3,3)$. Its connectivity, bipartition, Sombor matrix, product $B^{T}B$, singular values, complete spectrum and energy $64$ were independently recomputed within the AI conversation.
The result has not yet been checked by an independent graph-theory expert, peer reviewer or formal proof assistant. A literature search located the original conjecture, the 2023 partial-results paper and its 2024 corrigendum, but did not locate an equivalent connected counterexample. This search does not establish absolute priority, and no claim is made that this is the first or a new counterexample.
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
The source status is candidate and must not be represented as solved.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.