Uniqueness of currents in infinite resistive networks

If an infinite resistive network, whose edges have resistance 1ohm, satisfies a certain graph theoretical condition, then the homogeneous Kirchhoff equations have no nonzero solutions vanishing at infinity. Every vertex transitive graph with polynomial growth satisfies such a condition. Furthermore uniqueness holds in Cartesian products of infinite regular graphs. Graphs with more than one end and satisfying an isoperimetric inequality provide a counterexample to uniqueness. These results extend partially also to networks with nonconstant resistances.