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Can someone tell me or prove the formula for number of simple unlabelled undirected connected graphs possible on n vertices ?

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    The ever-helpful OEIS to the rescue: [A001349](http://oeis.org/A001349) – 2017-01-02
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    Always, include your attempts about that query. – 2017-01-02

2 Answers 2

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There is no closed formula known for this. You can see the first few terms of the sequence at OEIS, but that's about it.

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As wiki defined: A simple graph, as opposed to a multigraph, is an undirected graph in which both multiple edges and loops are disallowed.

There are $n(n-1)/2$ edges maximum edges in undirected graph because order doesn't matter.

Now, total number of possible such graph are $2^{n(n-1)/2}$, because each edge has two possibility either present in the graph are absent.

Similarly, for directed graph :

There are $n(n-1)$ edges maximum edges in undirected graph because order does matter.

Now, total number of possible such graph are $2^{n(n-1)}$, because each edge has two possibility either present in the graph are absent.

Read: http://compalg.inf.elte.hu/~tony/Oktatas/TDK/FINAL/Chap%204.PDF

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    Note that these are counts for *labeled* graphs (including ones that aren't connected). – 2017-01-02
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    I tried to come up with an answer but couldn't because of the vertices being unlabelled. I already knew the formula which you gave above for labelled vertices – 2017-01-02