Harmonic maximum principle on a finite graph

A real function $h$ on a finite connected graph satisfying $Lh=0$ everywhere is constant. At a maximum graph vertex, the identity $\mathrm{deg}(u)h(u)={\sum }_{v\sim u}h(v)$ forces every graph neighbour to have the same maximum. Connectedness propagates it to every graph vertex. This proves uniqueness of voltage up to an additive constant, and identifies sums of oppositely directed occupation voltages.