Almost every graph is asymmetric?

Question

Here is a question:

If i choose at random an isomorphism class of graph(no loops, undirected) on n vertices(with uniform probability on the set of such isomorphism classes), is the probability that the resulting graph has trivial automorphism group, going to 1 as n goes to $+\infty$?

I guess yes, but i do not have a neat argument.

Thanks for any suggestion.

Edit 1: I have the same question restricted to the isomorphism classes of regular n graph(each vertex the same degree). Do the asymmetric one appear almost surely also in this restricted space of graphs?

Edit 2: in the comment it turns out that is a theorem of Erdős, the first question. I'm still interested in knowing if my question in Edit 1 has been already answered.

Answer 1

For $k$-regular graphs the answer is also yes, see the paper http://www.math.ucla.edu/~bsudakov/automorphism.pdf by Kim, Sudakov and Vu for the proof. Note we must exclude $k=0,1,2$ along with complementary cases.