Abstract
We regard a graph as an electrical network where every edge is a resistance of 1 ohm. We show that, for an infinite graph of linear growth, Kirchhoff's laws combined with the finite power condition ensures the existence and uniqueness of an electric current when a current generator is added. We also show that the effective resistance between any two adjacent vertices in an infinite edge-transitive d-regular graph of moderate growth is 2 d , a result which has been expected by electrical engineers, but has previously been proved rigorously only in special cases.
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