What automorphisms on a Markov Chain imply a uniform limiting distribution? Consider an irreducible aperiodic Markov chain $M$, modeled as a connected directed graph with weighted edges. The existence of certain (graph) automorphisms on this Markov chain imply various symmetries regarding the limiting distribution. What properties can be demanded of an automorphism or set of automorphisms on $M$, in order for the limiting distribution to be uniform?
This is perhaps a nearly trivial observation, but I couldn't think of another general property just of the automorphisms that would ensure the limiting distribution is uniform. 

If the automorphism group of the corresponding weighted directed graph is vertex-transitive, then the limiting distribution must be uniform, since then no vertex can be distinguished from any other.