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I've been thinking about this for a few days, but I haven't found a general solution yet. How many distinct simple, connected, undirected graphs are there of n labelled vertices? For example, there is one for n = 2 and there are four for n = 3. Thanks in advance!

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    What is a "simply connected" graph? A tree?2012-05-27
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    I count three for $n=3$.2012-05-27
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    @ChrisEagle I am asking for your definition, not for the number.2012-05-27
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    Not necessarily a tree—it can have cycles. That's also why there are four for n=3, rather than just three.2012-05-27
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    @Phira: why is my definition relevant? It's the OP we need to hear from.2012-05-27
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    @Chris: So does "simply connected" just mean "connected" then?2012-05-27
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    @ChrisEagle Sorry.2012-05-27
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    Ach, sorry! I meant "simple, connected," not "simply connected," which I don't think is a term in graph theory. That must be the source of confusion.2012-05-27
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    Rats. I thought he was asking about trees. Funny thing is, I'd been thinking of asking that exact same question. Depending on the results of this, I still might.2012-05-27
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    This is [OEIS sequence A001187](http://oeis.org/A001187). When looking for this sort of sequence, it's a good idea to first search OEIS, both by text search and by determining the first few terms and searching with them.2012-05-27
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    I have written an answer based on the word "labelled", but I am still unsure why the word "distinct" is used in the question.2012-05-27
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    See also [this interesting question](http://math.stackexchange.com/questions/68457/number-of-connected-graphs-on-labeled-vertices-counted-according-to-parity).2012-05-27
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    @Phira Thanks for the answer. Unfortunately, I'm just some kid without much of a mathematics background, so I don't quite get most of your explanation. The first line makes sense, but what was your thought process in introducing G(x) and C(x), and what does the exp function do? Thanks!2012-05-27
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    @Chris Do you understand the symbols I have written? As in: Do you know exp and the sigma notation for sums?2012-05-27
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    @Phira I'm perfectly comfortable with sigma notation, but not this exp thing.2012-05-27
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    @Chris exp is the exponential function. http://en.wikipedia.org/wiki/Exponential_function2012-05-27
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    Oh. Well, now I feel silly. Thank you very much for the answer.2012-05-27

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