A note on the Poincar\'e and Cheeger inequalities for simple random walk on a connected graph

In 1991, Persi Diaconis and Daniel Stroock obtained two canonical path bounds on the second largest eigenvalue for simple random walk on a connected graph, the Poincar\'e and Cheeger bounds, and they raised the question as to whether the Poincar\'e bound is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincar\'e, establish some sufficient conditions on the canonical paths for the Poincar\'e bound to triumph, and show that there is always a choice of paths for which this happens.