  The entropy of random graph ensembles has gained widespread attention in the
field of graph theory and network science. We consider microcanonical ensembles
of simple graphs with prescribed degree sequences. We demonstrate that the
mean-field approximations of the generating function using the
Chebyshev-Hermite polynomials provide estimates for the entropy of finite-graph
ensembles. Our estimate reproduces the Bender-Canfield formula in the limit of
large graphs.
