Can one find all the subgroups of $S_n$ that are automorphism group of a graph with $n$ vertices?
I know that every subgroup of $S_n$ can be the automorphism group for a graph with enough vertices via Cayley's graphs. But I can't find anything imposing the restriction that the graph must have $n$ vertices.
I know that the result is false in general, e.g. there's no graph with 3 vertices such that $Aut(G)\cong C_3$ the $3$-cycle.
I was thinking that maybe one can find conditions on the generators of the automorphism group of a graph, so maybe one can find a familiy of subgroups of $S_n$ that have the property or something like that.