Suppose that there are $n$ vertices, we want to construct a regular graph with degree $p$, which, of course, is less than $n$. My question is how many possible such graphs can we get?

McKay and Wormald conjectured that the number of simple $d$regular graphs of order $n$ is asymptotically $$\sqrt 2 e^{1/4} (\lambda^\lambda(1\lambda)^{1\lambda})^{\binom n2}\binom{n1}{d}^n,$$ where $\lambda=d/(n1)$ and $d=d(n)$ is any integer function of $n$ with $1\le d\le n2$ and $dn$ even. Bender and Canfield, and independently Wormald, proved this for bounded $d$ in 1978, and Bollobás extended this to $d=O(\sqrt{\log n})$ in 1980. McKay and Wormald proved the conjecture in 19901991 for $\min\{d,nd\}=o(n^{1/2})$ [1], and $\min\{d,nd\}>cn/\log n$ for constant $c>2/3$ [2]. These remain the best results. The gap between these ranges remains unproved, though the computer says the conjecture is surely true there too. The formula apart from the $\sqrt2e^{1/4}$ has a simple combinatorial interpretation, and the universality of the constant $\sqrt2e^{1/4}$ is an enigma crying out for an explanation. Incidentally this conjecture is for labelled regular graphs. For isomorphism classes, divide by $n!$ for $3\le d\le n4$, since in that range almost all regular graphs have trivial automorphism groups (references on request). For $d=0,1,2,n3,n2,n1$, this isn't true. [1] Combinatorica, 11 (1991) 369382. http://cs.anu.edu.au/~bdm/papers/nickcount.pdf [2] European J. Combin., 11 (1990) 565580. http://cs.anu.edu.au/~bdm/papers/highdeg.pdf 


There is no closed formula (that anyone knows of), but there are asymptotic results, due to Bollobas, see A probabilistic proof of an asymptotic formula for the number of labelled regular graphs (1980) by B Bollobás (European Journal of Combinatorics) or Random Graphs (by the selfsame Bollobas). 


Looking up OEIS, some related sequences are A005176 for the number of nonisomorphic regular graphs on $n$ vertices, and A005177 for the number nonisomorphic connected regular graphs on $n$ vertices. 


And 'of course', if you really want those graphs you might have a look at genreg by Markus Meringer: http://www.mathe2.unibayreuth.de/markus/reggraphs.html 

