Number of different Graphs with n vertices

Asked Jan 03 '16 at 21:22

Active Jan 03 '16 at 21:34

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A graph has 7 vertices and 7 edges. How many different graphs (non- isomorphic) are possible?

The resulting graph has no constraints. i.e. it can be either connected or disconnected graph.

How can I extend this idea to graph with n vertices and 7 edges?

edited Jan 03 '16 at 21:34

asked Jan 03 '16 at 21:22

likecs

By "different", do you mean "nonisomorphic", or literally different? – Ravi Fernando Jan 03 '16 at 21:31
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question made more clear. – likecs Jan 03 '16 at 21:33
This kind of question occurs quite a lot - were none of the other answers relevant? Such as this one : http://math.stackexchange.com/questions/1072726/counting-simple-connected-labeled-graphs-with-n-vertices-and-k-edges?rq=1 – gilleain Jan 03 '16 at 21:52
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@gilleain: I haven't looked around for other answers, but that one appears to be for connected graphs. Of course, if you can find the answer for connected graphs (for all $N$ and $K \leq 7$), then disjoint unions will give you the desired graphs. – Ravi Fernando Jan 03 '16 at 22:02
I should add that I'm not an expert in this area, and it's possible that there's a nice closed-form answer I don't know of using (e.g.) generating functions. – Ravi Fernando Jan 03 '16 at 22:04
The following MSE link may prove relevant. – Marko Riedel Jan 03 '16 at 22:10
Do you consider graphs with multiple edges and loops? – M.U. Jan 04 '16 at 20:49
Self loops are allowed but not multiple edges. – likecs Jan 04 '16 at 21:30

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