Remarks on curvature dimension conditions on graphs

We show a connection between the $CD{E}^{\mathrm{\prime }}$ inequality and the $CD\psi$ inequality. In particular, we introduce a $C{D}_{\psi }^{\phi }$ inequality as a slight generalization of $CD\psi$ which turns out to be equivalent to $CD{E}^{\mathrm{\prime }}$ with appropriate choices of $\phi$ and $\psi$. We use this to prove that the $CD{E}^{\mathrm{\prime }}$ inequality implies the classical $CD$ inequality on graphs, and that the $CD{E}^{\mathrm{\prime }}$ inequality with curvature bound zero holds on Ricci-flat graphs.