Fastest Expected Time to Mixing for a Markov Chain on a Directed Graph

For an irreducible stochastic matrix T , the Kemeny constant K(T ) measures the expected time to mixing of the Markov chain corresponding to T . Given a strongly connected directed graph D, we consider the set ΣD of stochastic matrices whose directed graph is subordinate to D, and compute the minimum value of K, taken over the set ΣD. The matrices attaining that minimum are also characterised, thus yielding a description of the transition matrices in ΣD that minimise the expected time to mixing. We prove that K(T ) is bounded from above as T ranges over the irreducible members of ΣD if and only if D is an intercyclic directed graph, and in the case that D is intercyclic, we find the maximum value of K on the set ΣD. Throughout, our results are established using a mix of analytic and combinatorial techniques.