Entropy of microcanonical finite-graph ensembles

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.