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

Solution

Thomson principle states that the effective resistance is the minimum energy of a unit flow from

a

z

R_{eff} (a, z) = in f_{θ} \sum_{e} \frac{θ ( 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

Π_{k}

1 = (\sum_{e \in Π_{k}} θ (e))^{2} \leq (\sum_{e \in Π_{k}} \frac{θ ( e ) ^{2}}{c ( e )}) (\sum_{e \in Π_{k}} c (e)) .

For disjoint cutsets, summing these energy lower bounds and applying Thomson's principle gives the Nash-Williams inequality

R_{eff} (a, z) \geq \sum_{k = 1}^{m} (\sum_{e \in Π_{k}} c (e))^{- 1} .

E_{a} T_{z} + E_{z} T_{a} = 2 (\sum_{e \in E} c (e)) R_{eff} (a, z) .

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