Spectra of signed adjacency matrices

A signed adjacency matrix is a {−1, 0, 1}-matrix A obtained from the adjacency matrix A of a simple graph G by symmetrically replacing some of the 1’s of A by −1’s. Bilu and Linial have conjectured that if G is k-regular, then some A has spectral radius ρ(A) ≤ 2 √ k − 1. If their conjecture were true then, for each fixed k > 2, it would immediately guarantee the existence of infinite families of k-regular Ramanujan graphs, that is, families of graphs whose adjacency eigenvalues have second largest modulus at most 2 √ k − 1. This note presents some spectral properties that arose during an examination of the conjecture.