Gradient estimates on graphs with the $CD\psi (n,-K)$condition

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