I asked this question in stackexchange a few days back (https://math.stackexchange.com/questions/2741806/bounding-the-number-of-non-isomorphic-graphs-having-m-edges-and-no-isolated-ve?noredirect=1#comment5658566_2741806), but did not get any satisfactory answer there. So I decided to ask it again in Mathoverflow.

**Given a positive integer $m$, let $A_m$ denote the number of non-isomorphic graphs (connected or disconnected) having exactly $m$ edges and no isolated vertices. What is the tightest known bound on $A_m$?**

More specifically, is it true that $A_m \leq m^{m^\alpha}$ for some $\alpha < 1$?

I would be glad if someone points out that this is not the case. Counterexamples of the form $A_m \geq m^m$ or $A_m \geq m!$ will also be highly valued by me!