Bijective coding of general graphs (like Prüfer code but not for trees)

As pointed out by @EthanBolker, the problem of describing all graphs upto isomorphism is not even known to be NP-hard. But, in case you want to just enumerate the number of graphs of order $n$, then as the graphs are nothing but binary non-reflexive, symmetric relations on a set of $n$ elements, their number is $2^{frac{n(n-1)}{2}}$. As for directed graphs, since these needd not be symmetric, therefore, we have their number of $n$ vertices equals $2^{n(n-1)}$. Also see here