An Improved Upper Bound for the Bilu-Linial Conjecture via Interlacing Families

The Bilu-Linial conjecture asserts that every $d$-regular graph admits a signing $\sigma$ such that the spectral radius of the signed adjacency matrix $A_{\sigma }$ satisfies $\rho (A_{\sigma })\le 2\sqrt{d-1}$. Bilu and Linial also proved the weaker bound $O(\sqrt{d{\log }^{3}d})$ for graphs of maximum degree $d$. Marcus, Spielman, and Srivastava confirmed the conjecture in the case of $d$-regular bipartite graphs. In this paper, we prove that every graph of maximum degree $d$ has a signing $\sigma$ such that $$\rho (A_{\sigma })\le 2\sqrt{3(d-1)}\mathrm{.}$$ This removes the polylogarithmic factor from the estimate of Bilu and Linial and gives an explicit $2\sqrt{3(d-1)}$ two-sided spectral bound. The proof builds on the method of interlacing polynomials introduced by Marcus, Spielman, and Srivastava, together with results on mixed characteristic polynomials established by Marcus, Spielman, and Srivastava and by Bownik.