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

Abstract

We construct a finite connected simple cubic graph $F$ such that every signing of its edges yields a signed adjacency matrix with an eigenvalue outside $[-2\sqrt2,2\sqrt2]$. This disproves the Bilu--Linial signing conjecture for general regular graphs. The graph $F$ is not Ramanujan, and the conjecture restricted to Ramanujan base graphs remains open.