An algorithm to evaluate the spectral expansion

Abstract: 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.