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 atmost $1-c/n{\tau'}$, 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.