On a finite vertex set the graph notions agree, and on connected graphs the two path distances agree

Given: The hypotheses of the Statement.

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).

$E\subseteq [V{]}^2:=\{\{u,v\}\subseteq V:u\mathrm{\ne }v\}\mathrm{.}$ (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets).

A walk of length $\mathrm{\ell }$ is a finite vertex list (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

Vertices $u$ and $v$ of a graph $G$ are connected, or reachable from one another, when a path in $G$ has endpoints $u$ and $v$. Equivalently, a walk joins them, because turns such a walk into a path. (Connected graphs and connected components defined by the existence of vertex paths).

For vertices $u,v$ in one connected component, the published graph distance is the least length of a path joining them (Graph distance within a component, eccentricity, diameter and girth, including the acyclic convention).

These are the metric axioms, so shortest-path distance is a metric on $C$. (Shortest-path distance is a metric on every connected component).

By deleting zero or more closed segments from its vertex list, one obtains a path from $v_0$ to $v_{\mathrm{\ell }}$ of length at most $\mathrm{\ell }$. If $W$ repeats a vertex, the resulting path can be chosen to have length strictly less than $\mathrm{\ell }$. (Every walk between two vertices contains a path between the same endpoints).

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).

Every walk in a simple graph contains a path with the same endpoints and of no greater length (Every walk contains a path between the same endpoints, of no greater length).