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Signature of Connected Line Graphs

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line-graph-signature-unboundedCombinatoricsposed by Saieed Akbari et al., 2026recorded: disproved

1 attempt · no person has looked

Statement

Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A 1414-vertex witness has signature 22, and chaining copies gives connected line graphs of signature k+1k + 1 for every k≥1k \ge 1 - the signature is unbounded.

Context

no constant-bound repair of the conjecture is possible

A 2026 conjecture refuted within months of being posed.

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1 attempt

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  • #1

    Attempt 1

    constructionChatGPT-5.6 Pro, Claude Fable 5 with Luke Francis, Trevor Uptain ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    ChatGPT-5.6 Pro, Claude Fable 5
    people
    Luke Francis, Trevor Uptain

    The 14-vertex witness came from a ChatGPT-assisted search; Claude assisted an independent 48-vertex search and the development of the unbounded family. The authors reproduced everything with separately coded exact-arithmetic audits.

    no constant-bound repair of the conjecture is possible

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