Assume that $X$ is a connected $(q+1)$-regular undirected graph of finite order $n$. Let $A$ denote the adjacency matrix of $X$. Let $\lambda_1=q+1>\lambda_2\geq \lambda_3\geq \ldots \geq \lambda_n$ denote the eigenvalues of $A$. The spectral expansion of $X$ is defined by $$ \Delta(X)=q+1-\max_{2\leq i\leq n}|\lambda_i|. $$ By Alon--Boppana theorem, when $n$ is sufficiently large, $\Delta(X)$ is quite high if $$ \mu(X)=q^{-\frac{1}{2}} \max_{2\leq i\leq n}|\lambda_i| $$ is close to $2$. In this paper we introduce a number sequence $\{H_k\}_{k=1}^\infty$ and study its connection with $\mu(X)$. Furthermore, with the inputs $A$ and a real number $\varepsilon>0$ we design an algorithm to estimate if $\mu(X)\leq 2+\varepsilon$ in $O(n^\omega \log \log_{1+\varepsilon} n )$ time, where $\omega<2.3729$ is the exponent of matrix multiplication.

中文翻译：

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一种评估谱扩展的算法

假设$X$ 是有限阶$n$ 的连通$(q+1)$-正则无向图。让$A$表示$X$的邻接矩阵。令 $\lambda_1=q+1>\lambda_2\geq \lambda_3\geq \ldots \geq \lambda_n$ 表示 $A$ 的特征值。$X$ 的谱扩展定义为 $$ \Delta(X)=q+1-\max_{2\leq i\leq n}|\lambda_i|。$$ 根据Alon--Boppana定理，当$n$足够大时，如果$$ \mu(X)=q^{-\frac{1}{2}} \ max_{2\leq i\leq n}|\lambda_i| $$ 接近 $2$。在本文中，我们介绍了一个数列 $\{H_k\}_{k=1}^\infty$ 并研究了它与 $\mu(X)$ 的关系。此外，输入 $A$ 和实数 $\varepsilon>0$ 我们设计了一个算法来估计 $\mu(X)\leq 2+\varepsilon$ 在 $O(n^\omega \log \log_ {1+\varepsilon} n )$ 时间，其中 $\omega<2.3729$ 是矩阵乘法的指数。