How many different non-isomorphic random graphs of the Erdős–Rényi model exist?

Question

The Erdős–Rényi model $G(n,m)$ gives a random graph on n vertecies with m edges. I'm interested in the number of possible graphs, that can be generated that way.

If you ignore isomorphism, there are quite obvious $nCr(\frac{n(n-1)}{2}, m)$ possible graphs.

From my intuition these graphs should have lot's of isomorphic "copies", which are counted as well. So how many "truly diffrent" (not isomorphic to another) graphs do exist for $G(n.m)$?

I'm mainly interested in exact results (or bounds) for relative small n,m than in an asymptotic outcome, but if you have something regarding the later I'd happily look into it as well. — Simoris, Aug 05 '19 at 15:58

score 1 · Accepted Answer · answered Aug 06 '19 at 12:01

Since $G(n,m)$ may produce every graph on $n$ vertices and with $m$ edges, you seem to be simply asking about the number of non-isomorphic graphs with these numbers of vertices and edges. This sounds like a difficult counting problem, but it appears that there is, in fact, a (complicated) closed formula for this number, see this answer on MathOverflow. For concrete examples, see this sequence on OEIS.

How many different non-isomorphic random graphs of the Erdős–Rényi model exist?

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