  We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.
