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The Bilu-Linial signing conjecture for regular graphs

False for general regular graphs: an explicit cubic graph $F$ for which every signing produces an eigenvalue outside the Ramanujan interval. $F$ is not itself Ramanujan, and the conjecture restricted to Ramanujan base graphs - the form that would yield Ramanujan graphs of every degree by iteration - remains open.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 14, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Bilu-Linial signing conjecture for regular graphs · Bilu-Linial signing conjecture

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

ChatGPT
model · ai model contributor · OpenAI

Zhiqiang Xu
human · human collaborator

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VibeMathed record: The Bilu-Linial signing conjecture for regular graphs

specialist registry · passed

Checked here on 22 September 2026 against arXiv:2609.15591: the abstract gives a finite connected simple cubic graph $F$ every signing of which has an eigenvalue outside $[-2\sqrt2, 2\sqrt2]$ , and states the limitation plainly - $F$ is not Ramanujan, so the conjecture restricted to Ramanujan base graphs remains open. The reference list confirms Bilu and Linial (CPC 13 (2004) and Combinatorica 26 (2006)) and the Marcus-Spielman-Srivastava line. The mathematics was not checked here; eight days old, no referee.

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The Bilu-Linial signing conjecture for regular graphs

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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