On symmetric hollow integer matrices with eigenvalues bounded from below

Abstract

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, where $ρ$ is the unique real root of $x^3 = x + 1$. We show that for every $λ< λ^*$, there exists $n \in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-λ$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $λ\ge λ^*$.