Connected bounded-degree graphs satisfying CD(0,∞) for the unnormalised Laplacian are volume doubling and satisfy scale-invariant L² Poincaré inequalities with dilation two.
Delmotte,Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev
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Nonnegative Bakry--\'Emery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincar\'e Inequalities
Connected bounded-degree graphs satisfying CD(0,∞) for the unnormalised Laplacian are volume doubling and satisfy scale-invariant L² Poincaré inequalities with dilation two.