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

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... e 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.