Non-negative Ollivier curvature on graphs, reverse Poincaré inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates.

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has ‖ P t μ − P t ν ‖ 1 ≤ W 1 (μ , ν) t for all probability measures μ , ν where P t is the heat semigroup and W 1 is the ℓ 1 -Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the eigenvalue estimate λ 1 ≥ log ⁡ (2) / diam 2. [ABSTRACT FROM AUTHOR]