TITLE: Smallest non-zero eigenvalue of a (0,1) matrix
QUESTION [20 upvotes]: 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 $\mathbb{R}$)?  I would be interested in estimates or bounds as I imagine an exact answer is tricky.
I asked this question previously at https://math.stackexchange.com/questions/666493/smallest-non-zero-eigenvalue-of-a-0-1-matrix .

REPLY [20 votes]: The correct asymptotic behavour is $n^{-n/2(1+o(1))}$. This is proved in:
N. Alon and V. H. Vu, Anti-Hadamard matrices, coin weighing,
threshold gates and indecomposable hypergraphs, J. Combinatorial
Theory, Ser. A 79 (1997), 133-160.