arxiv:2609.15591

A 3-regular counterexample to the Bilu--Linial signing conjecture

Published on Sep 14

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Abstract

A finite connected simple cubic graph is constructed whose every edge signing produces an adjacency matrix eigenvalue outside a specified interval, refuting the Bilu–Linial signing conjecture for regular graphs.

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We construct a finite connected simple cubic graph F such that every signing of its edges yields an adjacency matrix with an eigenvalue outside [-2sqrt2,2sqrt2]. This disproves the Bilu--Linial signing conjecture for general regular graphs. The proof is elementary, using a four-vertex calculation and a scalar recurrence.

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