  In this paper we show that all nodes can be found optimally for almost all
random Erd\H{o}s-R\'enyi ${\mathcal G}(n,p)$ graphs using continuous-time
quantum spatial search procedure. This works for both adjacency and Laplacian
matrices, though under different conditions. The first one requires
$p=\omega(\log^8(n)/n)$, while the seconds requires $p\geq(1+\varepsilon)\log
(n)/n$, where $\varepsilon>0$. The proof was made by analyzing the convergence
of eigenvectors corresponding to outlying eigenvalues in the $\|\cdot\|_\infty
$ norm. At the same time for $p<(1-\varepsilon)\log(n)/n$, the property does
not hold for any matrix, due to the connectivity issues. Hence, our derivation
concerning Laplacian matrix is tight.
