  In this paper, we study the nonlinear $p$-Laplacian equation
  $$-\Delta_{p} u+V(x)|u|^{p-2}u=f(x,u) $$ with positive and periodic potential
$V$ on the lattice graph $\mathbb{Z}^{N}$, where $\Delta_{p}$ is the discrete
$p$-Laplacian, $p \in (1,\infty)$. The nonlinearity $f$ is also periodic in $x$
and satisfies the growth condition $|f(x,u)| \leq a(1+|u|^{q-1})$ for some $
q>p$. We first prove the equivalence of three function spaces on
$\mathbb{Z}^{N}$, which is quite different from the continuous case and allows
us to remove the restriction $q>p^{*}$ in [SW10], where $p^{*}$ is the critical
exponent for $ W^{1,p}(\Omega) \hookrightarrow L^{q}(\Omega)$ with $\Omega
\subset \mathbb{R}^{N}$ bounded. Then, using the method of Nehari [Neh60,
Neh61], we prove the existence of ground state solutions to the above equation.
