This paper investigates gradient estimates on graphs satisfying the $CD\psi(n,-K)$ condition with positive constants $n,K$, and concave $C^{ 1}$ functions $\psi:(0,+\infty)\rightarrow\mathbb{ R}$. Our study focuses on gradient estimates for positive solutions of the heat equation $\partial_{ t}u=\Delta u$. Additionally, the estimate is extended to a heat-type equation $\partial_{ t}u=\Delta u+cu^{ \sigma}$, where $\sigma$ is a constant and $c$ is a continuous function defined on $[0,+\infty)$. Furthermore, we utilize these estimates to derive heat kernel bounds and Harnack inequalities.
Categories: math.DG