  We prove that for combinatorial graphs with non-negative Ollivier curvature,
one has \[ \|P_t \mu - P_t \nu\|_1 \leq \frac{W_1(\mu,\nu)}{\sqrt{t}} \] for
all probability measures $\mu,\nu$ where $P_t$ is the heat semigroup and $W_1$
is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent
formulation of a version of reverse Poincar\'e inequality. Furthermore, this
estimate allows us to prove Buser inequality, Liouville property and the the
eigenvalue estimate $\lambda_1 \geq \log(2)/\operatorname{diam}^2$.
