= Discrete harmonic function
{title2=$Lh=0$}
{wiki}
= Harmonic function on a graph
{synonym}
A real function $h$ on the <vertices of a graph> of an unweighted <locally finite graph> is a <discrete harmonic function> at $u$ when $\deg(u)h(u)=\sum_{v\sim u}h(v)$, 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>.
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