For a graph G with n nodes and a non-trivial automorphism group, is there a way to tell how many other graphs with n nodes and the same automorphism groups exist?
Obviously most graphs will have a complement with the same automorphism group(except the graphs whose complement is the graph).
I know that for larger graphs most of them are asymmetric and those may have many other graphs with identical automorphism groups. I am pretty sure that the complete graph and the empty graph are the only two graphs with n nodes that have the symmetric group on n letters as their automorphism group.