Given a directed complete graph on $n$ vertices, is there an efficient algorithm for computing its automorphism group? Is there a nontrivial upper bound on the order of its automorphism group? How about the isomorphism of two such graphs? 



Automorphism group of directed complete graph
I'm guessing that by "directed complete graph" you want each edge directed in exactly one of the two possible ways. If so, you have a *tournament*. Automorphism and isomorphism for tournaments is potentially easier than for general directed graphs but nobody has proved that. One thing to note is that the automorphism group has odd order and is therefore solvable, which may facilitate group-theoretic approaches.


For an upper bound on the group size, take $n!$ and remove from it the highest power of 2, but probably that is way too large. A better bound would be the answer to this [stackoverflow question](https://math.stackexchange.com/questions/787581/largest-permutation-group-of-odd-order).


The best known algorithm needs time $n^{O(\log n)}$, which is better than the best known for general digraphs.


A paper which studies these questions is [here](http://link.springer.com/chapter/10.1007/978-3-540-74456-6_51). 


