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 $d\to \mathrm{\infty }$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm of eigenvectors of Erd\H{o}s-R'enyi random graph $G(n,p)$, answering a question raised by Dekel-Lee-Linial.