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 $$ \Delta(X)=\lambda_1-\max_{2\leq i\leq n}|\lambda_i|. $$ By the 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, 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.
翻译：假设美元X$是连接的美元(q+1),美元是固定定序的普通非方向图 美元。请用美元表示相邻基数 $X美元。如果美元足够大,则美元=Delta(X)=q ⁇ -farac {1}2\lambda_n美元=n美元。x美元的光谱扩展由美元=Delta(X) = lambda_1-max2\leq i\leq=leq n ⁇ lambda_i____i美元。如果美元足够大,那么美元=Delda_3\geq\geq\ geldots\geq\ geldots\geq\geq\geq\geq\ geldots\geq\geq 3qq@geq@gon $1\\\\\%2\ maxlqration 美元定义为$2美元。在本文中输入和数字为美元, 美元 美元-xl=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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