Harmonic functions of general graph Laplacians

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