Improved Bounds for the Bilu--Linial Conjecture via Spectral Recovery from Mixed Determinantal Polynomials

The Bilu--Linial conjecture asks whether every finite $d$-regular graph with $d \geq 2$ admits an edge signing $σ$ whose signed adjacency matrix $A_σ$ has spectral radius at most $2\sqrt{d-1}$. We prove that every signing meeting the mixed-root condition $r_{A_σ}\leq\sqrt{2(d-1)}$ satisfies \[ ρ(A