Gradient estimates and volume doubling for locally finite weighted graphs with $CD\psi (n,-K)$ condition

We study gradient estimates and volume growth on locally finite weighted graphs satisfying the $CD\psi (n,-K)$ condition with $K\ge 0$. We establish a variational inequality for the heat semigroup and derive from it a family of Li-Yau type gradient estimates. Furthermore, under suitable assumptions on $\psi$, we establish a uniform heat retention estimate for metric balls. Combined with a heat kernel Harnack inequality obtained from the established gradient estimate, this yields a curvature dependent exponential volume doubling estimate \begin{equation*} V(x,2r)\leq C e^{cKr^2}V(x,r). \end{equation*} When $K=0$, the result reduces to a uniform volume doubling and further implies that the bottom of the spectrum of $-\mathrm{\Delta }$ vanishes on infinite graphs.