  We present results on the concentration properties of the spectral norm
$\|A_p\|$ of the adjacency matrix $A_p$ of an Erd\H{o}s-R\'enyi random graph
$G(n,p)$. First we consider the Erd\H{o}s-R\'enyi random graph process and
prove that $\|A_p\|$ is uniformly concentrated over the range $p\in [C\log
n/n,1]$. The analysis is based on delocalization arguments, uniform laws of
large numbers, together with the entropy method to prove concentration
inequalities. As an application of our techniques we prove sharp sub-Gaussian
moment inequalities for $\|A_p\|$ for all $p\in [c\log^3n/n,1]$ that improve
the general bounds of Alon, Krivelevich, and Vu (2001) and some of the more
recent results of Erd\H{o}s et al. (2013). Both results are consistent with the
asymptotic result of F\"uredi and Koml\'os (1981) that holds for fixed $p$ as
$n\to \infty$.
