For a path $\gamma=(v_0,e_1,v_1,\ldots,e_k,v_k)$, its $\ell$-length is $\sum_{j=1}^k\ell(e_j)$. The corresponding <path metric> is
$$
\rho(x,y)=\inf_{\gamma:x\to y}\sum_{e\in\gamma}\ell(e).
$$
Because every edge length is at least one, $\rho(x,y)\geq1$ whenever $x\ne y$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 4 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-215/4/a/i/solution