标题： 正则图的任意谱边 显示英文标题

标题： Arbitrary Spectral Edge of Regular Graphs

摘要： 我们证明，对于每个$d\geq 3$和$k\geq 2$，序列的$d$-正则图的前$k$个特征值的极限点集是\[ \{(\mu_1,\dots,\mu_k): d=\mu_1\geq \dots\geq \mu_{k}\geq2\sqrt{d-1}\}. \]。Alon 和 Wei 得到了$k=2$的结果，我们的结果验证了他们的一个猜想。 我们的证明使用了一个从广义随机正则图分布中采样的无限随机图。 为了控制这个无限对象的谱行为，我们表明 Huang 和 Yau 对 Friedman 定理的证明（该定理界定了随机正则图的第二个特征值）可以推广到这个模型。 我们还界定了非回溯算子的迹，正如 Bordenave 在他独立证明 Friedman 定理时所做的那样。 显示英文摘要

摘要： We prove that for each $d\geq 3$ and $k\geq 2$, the set of limit points of the first $k$ eigenvalues of sequences of $d$-regular graphs is \[ \{(\mu_1,\dots,\mu_k): d=\mu_1\geq \dots\geq \mu_{k}\geq2\sqrt{d-1}\}. \] The result for $k=2$ was obtained by Alon and Wei, and our result confirms a conjecture of theirs. Our proof uses an infinite random graph sampled from a distribution that generalizes the random regular graph distribution. To control the spectral behavior of this infinite object, we show that Huang and Yau's proof of Friedman's theorem bounding the second eigenvalue of a random regular graph generalizes to this model. We also bound the trace of the non-backtracking operator, as was done in Bordenave's separate proof of Friedman's theorem.