We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is $O(log^2 n)$. This is a consequence of a more general result which shows that the spectral norm of the entrywise absolute value of the transfer impedance matrix of a graph is $O(log^2 n)$. This result implies a simple oblivious routing scheme based on electrical flows in the case of transitive graphs.
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