Remarks on curvature dimension conditions on graphs

We show a connection between the $CDE'$ inequality and the $CD\psi$ inequality. In particular, we introduce a $CD_\psi^\varphi$ inequality as a slight generalization of $CD\psi$ which turns out to be equivalent to $CDE'$ with appropriate choices of $\varphi$ and $\psi$. We use this to prove that the $CDE'$ inequality implies the classical $CD$ inequality on graphs, and that the $CDE'$ inequality with curvature bound zero holds on Ricci-flat graphs.