  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.
