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A Note on Ihara Zeta Function of Large Random Graphs

Research paper by O. Khorunzhiy

Indexed on: 04 Feb '16Published on: 04 Feb '16Published in: Mathematical Physics



Abstract

We consider the Ihara zeta function of the random Erdos-Renyi graphs with the edge probability $\rho/n$, where $n$ is the number of the graph vertices and study its asymptotic behavior in the limit when $n$ and $\rho$ tend to infinity. The Ihara formula relates the Ihara zeta function with the eigenvalue distribution of a normalized version of the discrete analog of the Laplace operator on graphs. We prove that the limiting eigenvalue distribution of the corresponding random matrix ensemble is given by a shift of the semi-circle distribution widely known in the spectral theory of random matrices.