Exact Mixing Times for Random Walks on Trees

We characterize the extremal structures for certain random walks on trees. Let G = (V,E) be a tree with stationary distribution π. For a vertex i ∈ V , let H(π, i) and H(i, π) denote the expected lengths of optimal stopping rules from π to i and from i to π, respectively. We show that among all trees with |V | = n, the quantities mini∈V H(π, i), maxi∈V H(π, i), maxi∈V H(i, π) and ∑ i∈V πiH(i, π) are all minimized uniquely by the star Sn = K1,n−1 and maximized uniquely by the path Pn.