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Asymmetric graph is a graph that has only trivial automorphism. Asymptotically, almost all finite graphs are asymmetric. I'm looking for upper bounds and lower bounds on the growth rate of the number of asymmetric graphs on n nodes.

Also, I'm looking for a function $f(n)$ that exactly counts the number of asymmetric graph on n nodes? Is this function efficiently computable?

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Sure it's computable: there are finitely many graphs on n nodes and a terminating algorithm for determining whether or not they are asymmetric. Whether it's efficiently computable... – Qiaochu Yuan Jan 31 '11 at 13:46
Thanks Qiaochu, I'll edit the question. – Mohammad Al-Turkistany Jan 31 '11 at 13:54
up vote 2 down vote accepted

You might look at the references in A003400 for possibly disconnected graphs and A124059 for connected ones. Neither gives a formula, even asymptotic.

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Thanks Ross, It seems that the growth rate must be bounded from below by exponential function. – Mohammad Al-Turkistany Jan 31 '11 at 15:48
It seems likely that you are right that most graphs are asymmetric. And on $n$ nodes there are $2^{n(n-1)/2}$ total graphs. – Ross Millikan Jan 31 '11 at 15:54
Thanks a lot Ross, I wonder why such a natural problem has not got enough attention. – Mohammad Al-Turkistany Apr 20 '11 at 17:08

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