For any set $ X$ set $ [X]^2 = \big\{\{x,y\}: x\neq y \in X\big\}$ . A simple undirected graph $ G=(V,E)$ is a pair of a set $ V$ with $ E\subseteq [V]^2$ . So the collection of all graphs on $ \{1,\ldots,n\}$ is $ {\cal O}(2^{n^2})$ .

Let us discard isomorphic graphs: Is then the number of pairwise non-isomorphic graphs on $ \{1,\ldots,n\}$ in $ {\cal O}(2^n)$ ? Or can a better bound be given?