Fix numbers n,k. Is there a closed formula known for the number of kregular graphs consisting of n edges? I have a method of enumerating kregular 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: http://www.mathe2.unibayreuth.de/markus/reggraphs.html 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. 

