For every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic $\mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$ vertices that is $\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d-1} + \epsilon$ (excluding the single trivial eigenvalue of~$d$).
翻译:对于每恒定值$d\geq 3美元和 $\epsilon > 0美元,我们给出一个确定值$\mathrm{poly}(n)-时间算法,在 $\ Theta(n) 美元顶点上输出一个固定的美元图,即 $\ epsilon$- near- Ramanujan;即其顶点值以2\ sqrt{d-1} + epsilon$(不包括 ~ d$ 的单小值) 。