Alon-Boppana定理的显式界
Explicit bounds from the Alon-Boppana theorem
摘要：
本文给出了具有给定第二特征值的有限$k$ -正则图的顶点数界的显式方法。设$X$是一个有限的$k$ -正则图，$μ_1(X)$是它的邻接矩阵的第二大特征值.从著名的Alon-Boppana定理得出，对于任何$ε> 0$，只有1000个这样的$X$与$μ_1(X) < (2 - ε) \sqrt{k - 1}$，我们有效地实现了Serre的定量版本的这个结果。对于任何$k$和$ε$，这给出了具有$μ_1(X) < (2 - ε) \sqrt{k - 1}$的$k$ -正则图中顶点数的显式上界。
英文摘要：
The purpose of this paper is to give explicit methods for bounding the number of vertices of finite $k$-regular graphs with given second eigenvalue. Let $X$ be a finite $k$-regular graph and $μ_1(X)$ the second largest eigenvalue of its adjacency matrix. It follows from the well-known Alon-Boppana Theorem, that for any $ε> 0$ there are only finitely many such $X$ with $μ_1(X) < (2 - ε) \sqrt{k - 1}$, and we effectively implement Serre's quantitative version of this result. For any $k$ and $ε$, this gives an explicit upper bound on the number of vertices in a $k$-regular graph with $μ_1(X) < (2 - ε) \sqrt{k - 1}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
