A note on the Poincaré and Cheeger inequalities for simple random walk on a connected graph

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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 Poincaré and Cheeger bounds, and they raised the question as to whether the Poincaré bound is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincaré, establish some sufficient conditions on the canonical paths for the Poincaré bound to triumph, and show that there is always a choice of paths for which this happens.