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Is there an explicit formula for the number of connected graphs with at most $n$ vertices?

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Unless I've misunderstood you, just just want to know how many distinct connected graphs there are on $n$ vertices. There is nothing simple known. OEIS gives values for up to $K_{20}$ and some links to references.

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    Is there an explict formula for the number of connected graphs with at most $n$ vertics?2012-07-14