An $(n,d,\lambda)$-graph is a $d$ regular graph on $n$ vertices in which the absolute value of any nontrivial eigenvalue is at most $\lambda$. For any constant $d \geq 3$, $\epsilon>0$ and all sufficiently large $n$ we show that there is a deterministic poly(n) time algorithm that outputs an $(n,d, \lambda)$-graph (on exactly $n$ vertices) with $\lambda \leq 2 \sqrt{d-1}+\epsilon$. For any $d=p+2$ with $p \equiv 1 \bmod 4$ prime and all sufficiently large $n$, we describe a strongly explicit construction of an $(n,d, \lambda)$-graph (on exactly $n$ vertices) with $\lambda \leq \sqrt {2(d-1)} + \sqrt{d-2} +o(1) (< (1+\sqrt 2) \sqrt {d-1}+o(1))$, with the $o(1)$ term tending to $0$ as $n$ tends to infinity. For every $\epsilon >0$, $d>d_0(\epsilon)$ and $n>n_0(d,\epsilon)$ we present a strongly explicit construction of an $(m,d,\lambda)$-graph with $\lambda < (2+\epsilon) \sqrt d$ and $m=n+o(n)$. All constructions are obtained by starting with known ones of Ramanujan or nearly Ramanujan graphs, modifying or packing them in an appropriate way. The spectral analysis relies on the delocalization of eigenvectors of regular graphs in cycle-free neighborhoods.

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各种程度和大小的显式扩展器

$(n,d,\lambda)$-graph 是 $n$ 顶点上的 $d$ 正则图，其中任何非平凡特征值的绝对值至多为 $\lambda$。对于任何常数 $d \geq 3$、$\epsilon>0$ 和所有足够大的 $n$，我们证明存在一个确定性的 poly(n) 时间算法，输出 $(n,d, \lambda)$-图（正好在 $n$ 个顶点上）与 $\lambda \leq 2 \sqrt{d-1}+\epsilon$。对于任何具有 $p \equiv 1 \bmod 4$ 素数和所有足够大的 $n$ 的 $d=p+2$，我们描述了 $(n,d, \lambda)$-graph（在正好 $n$ 个顶点）与 $\lambda \leq \sqrt {2(d-1)} + \sqrt{d-2} +o(1) (< (1+\sqrt 2) \sqrt {d-1 }+o(1))$，$o(1)$ 项趋向于 $0$，因为 $n$ 趋向于无穷大。对于每一个 $\epsilon >0$, $d>d_0(\epsilon)$ 和 $n>n_0(d,\epsilon)$ 我们提出了一个 $(m,d, \lambda)$-graph 与 $\lambda < (2+\epsilon) \sqrt d$ 和 $m=n+o(n)$。所有构造都是通过从已知的拉马努金图或近似拉马努金图开始，以适当的方式修改或打包它们而获得的。谱分析依赖于无环邻域中正则图的特征向量的离域化。