# Regularity of Erdos Renyi graph

I am interested to find out what’s the probability that an Erdos-Renyi graph $$\mathcal{G}(n,p)$$ is a regular graph?

I believe this is a really hard question whose non-asymptotic results are probably impossible to find. Therefore, I would probably settle for any asymptotic large deviation result or may be some upper bound of the probability.

I think there is no closed form of joint degree distribution of Erdos Renyi graph, which makes this a hard problem.

Any result in this direction will be very much appreciated.

1. Let $$G$$ and $$G'$$ be two labelled graphs on $$\{1,\ldots, n\}$$ [i.e., $$G=G'$$ iff $$ij \in E(G)$$ implies $$ij \in E(G')$$ and vice versa mere isomorphism doesn't count] with the same number $$m$$ of edges. If a graph is drawn according to ER$$(n,p)$$, the $$G$$ and $$G'$$ have the same probability $$P_n(m,p)$$ of being chosen (which is a function of $$n,p$$ and $$m=|E(G)| = |E(G')|$$). This I know can be calculated explicitly.
2. There is a formula (I think) that approximates the number $$N(k)$$ of $$k$$-regular graphs on $$\{1,\ldots, n\}$$.
So the probability $$P$$ of chossing a $$k$$-regular graph if drawing according to ER$$(n,p)$$ would come to
$$P= \sum_{k=1}^{n-1} N(k)P_n\left(\frac{kn}{2},p\right)$$