Getting Lost Efficiently

We consider the mixing time of random walks on a graph G, using the eigenvalue gap to measure the mixing time. We attempt to bound the mixing time in terms of the diameter Diam of G. We show that, for the standard random walk, there exists a family of graphs with diameter 3 but mixing time Ω(n), and a family of trees with diameter 3 but mixing time Ω(n). The main result is a scheme for re-weighting the edges of any graph so that the mixing time becomes O(Diam).