Remarks on Li-Yau inequality on graphs

In this paper, we study Li-Yau gradient estimates for the solutions $u$ to the heat equation ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u$ on graphs under the curvature condition $CD(n,-K)$ introduced by Bauer et al. in \cite{BHLLMY}. As applications, we derive Harnack inequalities and heat kernel estimates on graphs. Also we present a type of Hamilton gradient estimates.