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I am aware that the number of simple graphs possible with $n$ labelled vertices is $2^{n(n-1)/2}$. Is there any closed formula for the same problem with $n$ unlabelled vertices? In case of unlabelled vertices we have to eliminate graphs that are isomorphic.

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You can find a formula for the counting polynomial using Polya’s Enumeration Theorem as explained in $(3)$ of http://mathworld.wolfram.com/SimpleGraph.html

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