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 $\hat{I}{»}^{\ast }=\ddot{I}{}^{1/2}+\ddot{I}{}^{-1/2}\approx 2.01980$, where $\ddot{I}$ is the unique real root of $x^3=x+1$. We show that for every $\hat{I}»<\hat{I}{»}^{\ast }$, there exists $n\in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-\hat{I}»$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $\hat{I}»\ge \hat{I}{»}^{\ast }$.

8 pages