  Let $G$ be a $d$-regular graph $G$ on $n$ vertices. Suppose that the
adjacency matrix of $G$ is such that the eigenvalue $\lambda$ which is second
largest in absolute value satisfies $\lambda=o(d)$. Let $G_p$ with
$p=\frac{\alpha}{d}$ be obtained from $G$ by including each edge of $G$
independently with probability $p$. We show that if $\alpha<1$ then whp the
maximum component size of $G_p$ is $O(\log n)$ and if $\alpha>1$ then $G_p$
contains a unique giant component of size $\Omega(n)$, with all other
components of size $O(\log n)$.
