Summary: Let \(G(V, E)\) be a connected finite graph satisfying the \(CD_p^{\sqrt{\cdot}}(m, K)\) condition for \(p \geq 2\), \(m > 0\), \(K \leq 0\). In this paper we consider the elliptic gradient estimate for the solutions to the equation \[\Delta_p u = -\lambda_p |u|^{p-2}u\] on \(G\), where \(\Delta_p\) is the \(p\)-Laplace operator. As an application, we derive a lower bound estimate for the first nonzero eigenvalue of \(\Delta_p\) on \(G\).