An algorithm to evaluate the spectral expansion

Assume that $X$ is a connected $(q+1)$-regular undirected graph of finite order $n$. Let $A$ denote the adjacency matrix of $X$. Let $λ_1=q+1>λ_2\geq λ_3\geq \ldots \geq λ_n$ denote the eigenvalues of $A$. The spectral expansion of $X$ is defined by $$ Δ(X)=λ_1-\max_{2\leq i\leq n}|λ_i|. $$ By the Alon--Boppana theorem, when $n$ is sufficiently large, $Δ(X)$ is quite high if $$ μ(X)=q^{-\frac{1}{2}} \max_{2\leq i\leq n}|λ_i| $$ is close to $2$. In this paper, with the inputs $A$ and a real number $\varepsilon>0$ we design an algorithm to estimate if $μ(X)\leq 2+\varepsilon$ in $O(n^ω\log \log_{1+\varepsilon} n )$ time, where $ω<2.3729$ is the exponent of matrix multiplication.