A bijection of vertex sets is an isometry for the path metrics if and only if it is a graph isomorphism

Given: Connected simple graphs $G,H$ and a bijection $\phi :V(G)\to V(H)$.

A simple graph is a pair $(V,E)$ with $V$ any set and $E$ a set of two-element subsets of $V$ (Simple graphs on an arbitrary vertex set).

A walk of length $\mathrm{\ell }$ in a simple graph is a finite vertex list $(v_0,\dots ,v_{\mathrm{\ell }})$ with consecutive vertices adjacent; a path is a walk with distinct vertices; the graph is connected when it is nonempty and every two vertices are joined by a path (Walks, paths, connectedness and components in a simple graph on an arbitrary vertex set).

The path metric of a connected simple graph assigns to two vertices the least length of a path joining them (The path metric of a connected simple graph).

The path metric of a connected simple graph is a metric on its vertex set (The path metric of a connected simple graph is a metric on its vertex set).

A function $f:X\to Y$ is an isometric embedding when it preserves every distance, and it is an isometry when it is also bijective (Isometry, isometric embedding, and the subspace metric on a subset).

A graph isomorphism is a vertex bijection $\phi :V\to W$ such that $\{u,v\}$ is an edge exactly when $\{\phi (u),\phi (v)\}$ is an edge, for all distinct $u,v$ (Graph isomorphisms, automorphisms and graph complements).