Regular Erdos Renyi Graph

Question

For $G(n,1/2)$, the Erdos-Renyi graph on $n$ vertices where each edge appears with probability $1/2$. What is the probability (asymptotics as n tends to infinity) that the resulting graph is $n/2$-regular (assuming $n/2$ is an integer)? I am just looking for some rough bounds. In general, what is the probability that $G(n,p)$ is $d$ regular?

Answer 1

This question requires counting $d$ regular graphs, which apparently is known asymptotically for various $d$.

https://mathoverflow.net/questions/77730/how-many-p-regular-graphs-with-n-vertices-are-there

When you know the count, you can just multiply by $p^{nd/2} (1-p)^{n^2 - nd/2}$, which gives you the probability that you want.

(Judging by that MO post, I would guess that the probability you are looking for is not a simple one)