  The joint degree matrix of a graph gives the number of edges between vertices
of degree i and degree j for every pair (i,j). One can perform restricted swap
operations to transform a graph into another with the same joint degree matrix.
We prove that the space of all realizations of a given joint degree matrix over
a fixed vertex set is connected via these restricted swap operations. This was
claimed before, but there is an error in the previous proof, which we
illustrate by example. We also give a simplified proof of the necessary and
sufficient conditions for a matrix to be a joint degree matrix. Finally, we
address some of the issues concerning the mixing time of the corresponding MCMC
method to sample uniformly from these realizations.
