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For a finite connected undirected electrical network with positive edge electrical conductances $c_e$, the Thomson principle stateswhere $a\ne z$, each undirected edge is counted once with an arbitrary orientation, and $\theta$ ranges over unit flows with divergence $+1$ at $a$, $-1$ at $z$, and zero elsewhere. The expression is the dissipated energy for unit electric current.

To prove the minimum, let $i$ be the current flow arising from a voltage $u$ that drives one unit of electric current. Write another unit flow as $i+h$, so $h$ has zero divergence. Since $i(e)/c_e$ is the endpoint voltage difference, summation by parts givesHence the energy of $i+h$ is the energy of $i$ plus the nonnegative energy of $h$. The energy of $i$ is $u(a)-u(z)=R_{\mathrm{eff}}(a,z)$ by Ohm's law. For two resistors of electrical resistances $2$ and $3$ in series, every unit flow uses both, giving effective resistance $5$.

Walter L. Ruzzo coauthored bounds linking random walk cover time to electrical-network resistance. The paper also proves the exact commute time of a random walk identity, enabling deterministic effective resistance calculations to control expected exploration times.^[ref]