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It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most $2\sqrt{d-1} +o(1)$ with probability tending to one as the size of the graph tends to infinity.
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http://dlmf.nist.gov/, Release 1.0.18 of 2018-03-27
NIST Digital Library of Mathematical Functions · 2018
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