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Fix numbers n,k. Is there a closed formula known for the number of k-regular graphs consisting of n edges? I have a method of enumerating k-regular graphs on n edges, and would like to have a number to compare the algorithm against.

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I don't think there's any closed form formula, but you can compare your results with the data in OEIS, such as here: (for 3-regular graphs) or here: This is indexed by the number of vertices, not edges, but of course for fixed k this is the same up to a factor of k/2. –  Alon Amit Apr 21 '10 at 18:44

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I think the answer is no, but I would consult the following link:

which contains tables of the sums of the numbers you are interested in. The author is very generous with sharing data that is not posted online.

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