  We consider the following logarithmic Schr\"{o}dinger equation
  $$
  -\Delta u+h(x)u=u\log u^{2}
  $$ on a locally finite graph $G=(V,E)$, where $\Delta$ is a discrete
Laplacian operator on the graph, $h$ is the potential function. Different from
the classical methods in Euclidean space, we obtain the existence of global
solutions to the equation by using the variational method from local to global,
which is inspired by the works of Lin and Yang in \cite{LinYang}. In addition,
when the potential function $h$ is sign-changing, we prove that the equation
admits infinitely many solutions with high energy by using the symmetric
mountain pass theorem. We extend the classical results in Euclidean space to
discrete graphs.
