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.

Result: Disproved
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Spectral graph theory
Posed by: Nima Ghanbari
Year posed: 2021
Years open: 5y
Solved: 2026-07-30
Model: GPT-5.6 Thinking
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Significance: 5 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What the AI did

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.

Verification

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.

Source

Sombor Energy Conjecture Counterexample

Submitted by Lamp

Sombor-Energy Conjecture

What the AI did

Verification

Source

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