# Upper bound on the number of vertex transitive graphs

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

• $2^{n-1}$ times the number of minimal transitive groups would be a fair bound, but I don't know a bound on the number of minimal transitive groups. For this reason, adding a group category to your question might be a good idea. – Brendan McKay Jan 22 '14 at 8:30
• What is the reason for the $2^{n-1}$ factor? – Amit Levi Jan 27 '14 at 16:56
• There are $2^{n-1}$ ways to choose the neighbours of the first vertex. Then the group action gives all the other edges. – Brendan McKay Jan 27 '14 at 21:23

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 $d=3$.