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.
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.