Is there a known upper bound on the number of vertex transitive graphs on $n$ vertices?



Upper bound on the number of vertex transitive graphs
I think only very rough bounds are known.
In <http://arxiv.org/abs/1210.5736>, it is conjectured that the answer for fixed valency d is roughly of the form $n^{d\log n}$. 
This is proved for Cayley graphs and for $d=3$.


As Brendan mentions, obtaining upper bounds on the number of minimal transitive groups of a given degree would yield upper bounds. The best results in this direction are due to Pyber, I believe, but are not tight. I think this may be in :


Pyber, László
Asymptotic results for permutation groups. (English summary) Groups and computation (New Brunswick, NJ, 1991), 197–219, 
DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 11, Amer. Math. Soc., Providence, RI, 1993.


