# 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?