Li-Yau inequality under $CD(0,n)$ on graphs

We introduce a modified non-linear heat equation $\partial_t u = \Delta u + \Gamma u$asasubstituteof$\log P_t f$where$P_t$ is the heat semigroup. We prove an exponential decay of $\mathrm{\Gamma }u$ under the Bakry Emery curvature condition $CD(K,\mathrm{\infty })$ and prove the Li-Yau inequality $-\Delta u_t \leq \frac{n}{2t}$undertheBakryEmerycurvaturecondition$CD(0,n)$. From this, we deduce the volume doubling property which solves a major open problem in discrete Ricci curvature. As an application, we show that there exist no expander graphs satisfying $CD(0,n)$.