Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 3 / b / Solution

Let $N=|V_n|$. Since the degrees lie between $1$ and $\Delta$,The Generalized Cheeger inequality and ${\Phi }_{\ast }(u)\ge {\Phi }_{\ast }(c)\ge \alpha$ for $u\le c$ giveFor all larger $u$, part (a) gives $\Lambda (u)\ge 1/t_{\mathrm{rel}}$. Split the supplied spectral-profile integral at $c$ to obtain