Remarks on curvature dimension conditions on graphs

Abstract

We show a connection between the \(CDE'\) inequality introduced in Horn et al. (Volume doubling, Poincare inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the \(CD\psi \) inequality established in Munch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). 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.