A Proof of Alon's Second Eigenvalue Conjecture and Related Problems

A $d$-regular graph has largest or first (adjacency matrix) eigenvalue $/lambda 1=d$. Consider for an even $d/ge 4$, a random $d$-regular graph model formed from $d/2$ uniform, independent permutations on $/{1,/ldots,n/}$. The author shows that for any $/epsilon>0$ all eigenvalues aside from $/lambda 1=d$ are bounded by $2/sqrt{d-1}/;+/epsilon$ with probability $1-O(n{-/tau})$, where $/tau=/lceil /bigl(/sqrt{d-1}/;+1/bigr)/2 /rceil-1$. He also shows that this probability is at most $1-c/n{/tau'}$, for a constant $c$ and a $/tau'$ that is either $/tau$ or $/tau+1$ (""more often"" $/tau$ than $/tau+1$). He proves related theorems for other models of random graphs, including models with $d$ odd.