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 ${\lambda }^{⁎}={\rho }^{1/2}+{\rho }^{-1/2}\approx 2.01980$, where ρ is the unique real root of $x^3=x+1$. We show that for every $\lambda <{\lambda }^{⁎}$, there exists $n\in N$ 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 $\lambda \ge {\lambda }^{⁎}$.