On symmetric hollow integer matrices with eigenvalues bounded from below

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