 Discrete harmonic function

A real function $h$ on the vertices of a graph of an unweighted locally finite graph is a discrete harmonic function at $u$ when $\mathrm{deg}(u)h(u)={\sum }_{v\sim u}h(v)$, equivalently $Lh(u)=0$ for the Graph Laplacian. For positive degree this says its value is the average over its graph neighbours; at an isolated vertex of a graph the Laplacian condition is vacuous. On an electrical network the corresponding equation uses conductance weights and the Weighted graph Laplacian. Voltages away from sources and sinks, and hitting probabilities away from their prescribed boundaries, are discrete harmonic functions.