It seams that there is no exact formula. There are some results about asymptotic behavior. Start digging here "https://mathoverflow.net/questions/77730/how-many-p-regular-graphs-with-n-vertices-are-there"
This is a very good question. Actually some exact formulas can be obtained for some special classes of graphs and I think that regular and bi-regular graphs could be enumerated if you consider them by classes. What I do not know is how those classes can be chosen or considered. The algorithm designed in the first article helps us to determine the exact number of a special class of graphs called geodetic which are homeomorphic to complete graphs and also the Petersen graph. We know that the quality of work of a computer network can be significantly improved by changing the topology of connections (which in the case of homeomorphism can be done by knowing the exact number of graphs of given diameter d that represents the same network topology.) The second article deals with the problem of obtaining a network with optimum indices of network operation. The remaining two articles deal directly with the problem of finding exact and asymptotic formulas for the number of some classes of geodetic networks.
1.Article Geodetic Graphs Homeomorphic to a Given Geodetic Graph
2.Article Topological Analysis and Synthesis of Structures related to ...
3Article Enumeration of labeled geodetic graphs with small cyclomatic number
4.Article Enumeration of labeled geodetic planar graphs
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