  We provide the regularity and the Cohen-Macaulay type of binomial edge ideals
of Cohen-Macaulay cones, and we show the extremal Betti numbers of some classes
of Cohen-Macaulay binomial edge ideals: Cohen-Macaulay bipartite and fan
graphs. In addition, we compute the Hilbert-Poincar\'e series of the binomial
edge ideals of some Cohen-Macaulay bipartite graphs.
