Smallest non-zero eigenvalue of a (0,1) matrix What's the smallest absolute value possible of a non-zero eigenvalue of an $n$ by $n$ square matrix whose entries are either $0$ or $1$ (all operations are over $\mathbb{R}$)?

For a $2\times 2$ matrix, the smallest is $\left| \dfrac{1-\sqrt{5}}{2}\right|$ from $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ or $\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$.

For a $3\times 3$ matrix, the smallest is $\left|\frac{1}{2}(3-\sqrt{5})\right|$ from $\begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ or any other matrix with all entries $1$ except for a single off (main) diagonal $0$.

Best so far - have more calculations to run and check
For a $4\times 4$ matrix, the smallest is $\left|2-\sqrt{3}\right|$ from $\begin{pmatrix} 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}$
