We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an Liouville type theorem which is a quantitative integral estimate of harmonic functions analogous to Karp’s theorem for Riemannian manifolds. As corollaries we obtain Yau’s -Liouville type theorem on graphs, identify the domain of the generator of the semigroup on and get a criterion for recurrence. As a side product, we show an analogue of Yau’s Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces.
