Equivalent Properties of CD Inequality on Graph

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincar'e type inequalities and reverse Poincar'e inequalities. And we also obtain one equivalent property of gradient estimate for a new notion of curvature-dimension conditions $CD{E}^{\mathrm{\prime }}(\mathrm{\infty },K)$ at the same assumption of graphs.