Robust Isolated Vertices in a Random Graph

In the Erdos-Renyi random graph model $G(n,p)$, each of the $\binom{n}{2}$ possible edges among $n$ labeled vertices is included independently with probability $p$. A vertex is isolated if it has no edges. Find the expected number of isolated vertices.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.