  We are mainly concerned with the nonlinear $p$-Laplace equation
\begin{equation*}
  -\Delta_pu+\rho|u|^{p-2}u=\psi(x,u) \end{equation*} on a locally finite graph
$G=(V,E)$, where $p$ belongs to $(1, +\infty)$. We obtain existence of positive
solutions and positive ground state solutions by using the mountain-pass
theorem and the Nehari manifold respectively. Moreover, we also analyze the
asymptotic behavior for a sequence of positive ground state solutions. Compared
with all the existing relevant works, our results have made essential
improvements in at least three aspects: $(i)$ $p$ can take any value in $(1,
+\infty)$; $(ii)$ the conditions on the graph $G$ and the potential $\rho$ are
relaxed; $(iii)$ for the existence of positive solutions, the growth condition
in previous works on the nonlinear term $\psi(x,s)$ as $s\rightarrow +\infty$
is removed.
