Is anything known about the algebraic properties of random graphs? For example, what is the expected size of the automorphism group of $G(n,p)$? What is the probability that $G(n,p)$ is vertex-transitive?
I suspect that random graphs almost never have algebraically structure (i.e. $\mathbb{E} \vert \text{Aut } G(n,p) \vert \to 1$ as $n \to \infty$, $G(n,p)$ is almost never vertex-transitive, etc.), but google doesn't turn anything up.