If you are interested in the case for the discrete Laplacian, check out the paper of Rigoli, Salvatori, and Vignati titled "Liouville properties on graphs" (DOI: 10.1112/S0025579300012031).
Among the results proven is the following:
Let $G$ be a graph and let $q$ be an arbitrary point in $G$. Let $u$ be a $p$-subharmonic function on $G$ for $p > 1$. Suppose that for all $R$ sufficiently large $ \sup_{B_R(q)} u \lesssim \frac{(R\log R)^{(p-1)/p}}{|S_R(q)|^{1/p}} $ and $ |S_R(q)| \lesssim (R\log R)^{p-1} $ where $S_R(q) = B_R(q) \setminus B_{R-1}(q) $ is the "sphere of the radius $R$", then $u$ is constant.
The requirement on the volume growth rate of balls of radius $R$ is typical: this is not just the case for graphs. Liouville theorems for non-compact, complete Riemannian manifolds are usually proven under the assumption of a lower bound on the Ricci curvature, which can be used to prove volume growth bounds on the Riemannian manifold (the simplest example being the Bishop-Gromov theorem).