On graph theory Mertens' theorems

In this paper, we study graph-theoretic analogies of the Mertens' theorems by using basic properties of the Ihara zeta-function. One of our results is a refinement of a special case of the dynamical system Mertens' second theorem due to Sharp and Pollicott.Comment: 13 page

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 $\lambda_1=q+1>\lambda_2\geq \lambda_3\geq \ldots \geq \lambda_n$ 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