I am trying to count the number of $(d\_v,d\_c)$ regular bipartite graphs. To be specific, let $n,m,d\_v,d\_c$ be positive integers such that 
$$n\times d\_v=m\times d\_c.$$
Then, what is the number of bipartite graphs $\mathcal{G}=(L\cup R, E)$, where $L$ is the set of left vertices, $R$ is the set of right vertices with the property that each left vertex has $d\_v$ edges incident on it, and each right vertex has $d\_c$ edges incident on it (allowing parallel edges, i.e., there can be more than one edge for a given pair of vertices)? Can one find a general expression for this? 


This problem can be reformulated as a ball and bin problem in the following way: We have a total of $nd\_v$ balls, of $n$ colors, with $d\_v$ balls of each color. What is the number of ways of putting these balls in $m$ nonidentical bins so that each bin has exactly $d\_c$ balls? Let me denote this number $\psi(n,d\_v,d\_c)$ A quick thought about this gives me the following bounds:
$$
 \frac{(nd\_v)!}{\left(d\_v! \right)^n(d\_c!)^m}\leq\psi(n,d\_v,d\_c)\leq \frac{(nd\_v)!}{\left(d\_v! \right)^n}.
$$
Can one do better?


I think there must be some work that has addressed this problem but I have not been able to find anything. Any help would be appreciated.



Counting the number of $(d_v,d_c)$ regular bipartite graphs
There is no useful exact formula except for tiny $d\_v$ or $d\_c$. The best asymptotic results appear in [this paper of Canfield and McKay](http://arxiv.org/abs/math/0703600) for dense matrices and [this paper of Greenhill and McKay](http://arxiv.org/abs/0707.0340) for sparse matrices. There is an unsolved gap between the two ranges. The first paper has a conjecture that probably holds for all values of the parameters.


