  Consider the random graph sampled uniformly from the set of all simple graphs
with a given degree sequence. Under mild conditions on the degrees, we
establish a Large Deviation Principle (LDP) for these random graphs, viewed as
elements of the graphon space. As a corollary of our result, we obtain LDPs for
functionals continuous with respect to the cut metric, and obtain an asymptotic
enumeration formula for graphs with given degrees, subject to an additional
constraint on the value of a continuous functional. Our assumptions on the
degrees are identical to those of Chatterjee, Diaconis and Sly (2011), who
derived the almost sure graphon limit for these random graphs.
