  In this paper, we consider the existence and multiplicity of solutions for
the logarithmic Schr\"{o}dinger equation on lattice graphs $\mathbb{Z}^N$ $$
-\Delta u+V(x) u=u \log u^2, \quad x \in \mathbb{Z}^N, $$ When the potential
$V$ is coercive, we obtain infinitely many solutions by adapting some arguments
of the Fountain theorem. In the cases of periodic potential, asymptotically
periodic potential and bounded potential, we first investigate the existence of
ground state solutions via the variation methods, and then we generalize these
results from $\mathbb{Z}^N$ to quasi-transitive graphs. Finally, we extend the
main results of the paper to the $p$-Laplacian equation with the logarithmic
nonlinearity.
