What is the best known asymptotic formula for the number of graphs with a given degree sequence $(d_1, ... ,d_n)$, when the degrees are bounded by a constant and the number of vertices $n$ goes to infinity?

There are several papers of Mckay et al that provide bounds when the degrees are all $O(n^{\frac{1}{2}})$, but I am interested in bounded degrees, and I was hoping for more exact bounds for this special case. There are works on this as well, but they all seem to be pretty old, so I was wondering what the current state of the art is.