Sparse random graphs: Eigenvalues and Eigenvectors

In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\rightarrow \infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm of eigenvectors of Erdős-Rényi random graph $G(n,p)$, answering a question raised by Dekel-Lee-Linial.

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