The Best Mixing Time for Random Walks on Trees

We characterize the extremal structures for mixing walks on trees that start from the most advantageous vertex. Let G=(V,E) be a tree with stationary distribution π . For a vertex v ∈ V , let H(v,π ) denote the expected length of an optimal stopping rule from v to π . The best mixing time for G is min _v ∈ V H(v,π ) . We show that among all trees with |V|=n , the best mixing time is minimized uniquely by the star. For even n , the best mixing time is maximized by the uniquely path. Surprising, for odd n , the best mixing time is maximized uniquely by a path of length n-1 with a single leaf adjacent to one central vertex.