Is there a way to algorithmically find all graphs that have a certain automorphism group on $n$ vertices? For example, is there a way to find all graphs with automorphism group $A_5$ on $1000$ vertices? Obviously you could look at all graphs, and then compute the automorphism group of each graph, but this is very inefficient.
I would also be interested in a way to count the number of graphs with a given automorphism group, or to generate (or count) all graphs that have an automorphism group of a certain order. Special cases (e.g. trivial automorphism group) would be interesting as well.
Clearly the only graph on $n$ vertices with automorphism group $S_n$ is $K_n$.