Elliptic gradient estimate for the $p$−Laplace operator on the graph

Let $G(V,E)$ be a connected locally finite graph. In this paper we consider the elliptic gradient estimate for solutions to the equation\[\Delta_p u - \lambda_p {\lvert u \rvert}^{p-2} u\]on $G$ with the $\mathrm{CD}^\psi_p (m,-K)$ condition, where $p \geq 2$, $m \gt 0$, $K \geq 0$, and $\Delta_p$ denotes the $p\textrm{-}$Laplacian. As applications, we can derive Liouville theorems and the Harnack inequality.