Solution
= Solution
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.