An (n, d, λ)-graph is a d regular graph on n vertices in which the absolute value of any nontrivial eigenvalue is at most λ. For any constant d ≥ 3, ϵ > 0 and all sufficiently large n we show that there is a deterministic poly(n) time algorithm that outputs an (n, d, λ)-graph (on exactly n vertices) with \(\lambda \le 2\sqrt {d - 1} + \). For any d=p + 2 with p ≡ 1 mod 4 prime and all sufficiently large n, we describe a strongly explicit construction of an (n, d, λ)-graph (on exactly n vertices) with \(\lambda \le \sqrt {2\left({d - 1} \right)} + \sqrt {d - 2} + o\left(1 \right)\left({< \left({1 + \sqrt 2} \right)\sqrt {d - 1} + o\left(1 \right)} \right)\), with the o(1) term tending to 0 as n tends to infinity. For every ϵ> 0, d> d₀(ϵ) and n>n₀(d, ϵ) we present a strongly explicit construction of an (m, d, λ)-graph with \(\lambda<\left({2 +} \right)\sqrt d \) and m = n + o(n). All constructions are obtained by starting with known Ramanujan or nearly Ramanujan graphs and modifying or packing them in an appropriate way. The spectral analysis relies on the delocalization of eigenvectors of regular graphs in cycle-free neighborhoods.

（n，d，λ）图是n个顶点上的d正则图，其中任何非平凡特征值的绝对值最多为λ。对于任何常数d ≥3，ε > 0且所有足够大Ñ我们表明，有一个确定性的聚（n）的时间的算法，输出一个（N，d，λ） -图（上恰好Ñ顶点）与\（\ lambda \ le 2 \ sqrt {d-1} + \）。对于任何d = p + 2 p ≡1模4素和所有足够大Ñ，我们描述一个（有一种强烈的显式施工N，d，λ）-图（恰好在n个顶点上）具有\（\ lambda \ le \ sqrt {2 \ left（{d-1} \ right）} + \ sqrt {d -2} + o \ left（1 \ right ）\ left（{<\ left（{1 + \ sqrt 2} \ right）\ sqrt {d-1} + o \ left（1 \ right）} \ right）\），其中o（1）项趋向于n趋于无穷大，因此为0 。对于每一个ϵ > 0，d > d ₀（ϵ）和n > n ₀（d，ϵ），我们给出一个（m，d，λ）图的强显式构造，其中\（\ lambda <\ left（{ 2 +} \ right）\ sqrt d \）和m = n +o（n）。通过从已知的Ramanujan图或几乎Ramanujan图开始并以适当的方式对其进行修改或打包，可以获取所有构造。频谱分析依赖于无环邻域中正则图特征向量的去域化。