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Given positive edge lengths $\ell (e)$ on a graph, define the distance between two vertices bywhere $\gamma$ ranges over finite graph paths. An absent path gives distance $+\infty$. On a finite connected graph, this is a metric: reversing and concatenating routes gives symmetry and the triangle inequality, while the least edge length bounds every nonempty path away from zero. It is also a metric on a connected locally finite graph, since every path leaving a fixed vertex pays at least the positive minimum of its finitely many incident edge lengths.

Positive edge lengths alone do not guarantee a metric on an arbitrary infinite graph. Join two vertices $u,v$ through intermediate vertices $z_n$, with both edges $u{z}_n,z_{n}v$ having length $1/n$. Then $d(u,v)\le 2/n$ for every $n$, so distinct vertices have distance zero. On each finite-distance component the general construction is a pseudometric; its zero-distance classes give a metric quotient.