Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 4 / a / Solution

For stationary edge flow $Q(x,y)={\pi }^G(x)P(x,y)$, the bottleneck ratio isUsing $f=1_S$ in the variational characterization,Taking the infimum gives ${\gamma }^G\le 2{\Phi }_{\ast }^G$.