Vu’s conjecture on spectral determination for symmetric ±1 matrices

Establish that, for almost all n × n symmetric matrices with entries in {−1, +1}, the multiset of eigenvalues uniquely determines the matrix among symmetric ±1 matrices; equivalently, show that the probability a random symmetric ±1 matrix is determined by its spectrum tends to 1 as n → ∞.

Background

Following Haemers’ conjecture for graphs, Vu proposed an analogous statement in the setting of random symmetric ±1 matrices, seeking a high-probability spectral characterization in a non-graph, matrix ensemble context.

The present paper’s findings for digraphs suggest the importance of the real symmetric condition in such spectral determination phenomena, underscoring why this conjecture remains significant.