Poincaré inequalities on graphs

Abstract

Every graph of bounded degree endowed with the counting measure satisfies a local version of L^p-Poincaré inequality, p ∈ [1, ∞]. We show that on graphs which are trees the Poincaré constant grows at least exponentially with the radius of balls. On the other hand, we prove that, surprisingly, trees endowed with a flow measure support a global version of L^p-Poincaré inequality, despite the fact that they are nondoubling measures of exponential growth.