We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees.
In this note, we demonstrate an instance of bounded-degree graphs of size @math , for which the total variation mixing time for the random walk is decreased by a factor of @math if we multiply the edge-conductances by bounded factors in a certain way.