A proof of Alon's second eigenvalue conjecture and related problems

A d-regular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. In this paper we show the following conjecture of Alon. Fix an integer d > 2 and a real ε > 0. Then for sufficiently large n we have that "most" d-regular graphs on n vertices have all their eigenvalues except λ1 = d bounded above by 2√d-1 + ε. Our methods, being trace methods, also bound those eigenvalues below by -2√d-1 - ε.

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