<Thomson principle> states that the effective resistance is the minimum energy of a unit flow from $a$ to $z$:
$$
R_{\mathrm{eff}}(a,z)=\inf_\theta
\sum_e\frac{\theta(e)^2}{c(e)}.
$$
An edge cutset separating $a$ and $z$ is a set of edges whose removal disconnects them. Every unit flow has net flux one across each such cutset $\Pi_k$. By <Cauchy-Schwarz inequality>,
$$
1=\left(\sum_{e\in\Pi_k}\theta(e)\right)^2
\leq\left(\sum_{e\in\Pi_k}\frac{\theta(e)^2}{c(e)}\right)
\left(\sum_{e\in\Pi_k}c(e)\right).
$$
For disjoint cutsets, summing these energy lower bounds and applying Thomson's principle gives the <Nash-Williams inequality>
$$
R_{\mathrm{eff}}(a,z)
\geq\sum_{k=1}^m\left(\sum_{e\in\Pi_k}c(e)\right)^{-1}.
$$
The <commute time identity> is
$$
\mathbb E_aT_z+\mathbb E_zT_a
=2\left(\sum_{e\in E}c(e)\right)R_{\mathrm{eff}}(a,z).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 215 / 5 / a / Solution

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{wiki}

<Thomson principle> says that effective resistance is the minimum energy among unit flows from $a$ to $z$:
$$
R_{\mathrm{eff}}(a,z)=\inf_\theta\sum_e\frac{\theta(e)^2}{c(e)}.
$$

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