Graph distance within a component, eccentricity, diameter and girth, including the acyclic convention

Let $u$ and $v$ lie in the same connected component of a graph $G$. Their distance is

$d_G(u,v):=\min \{\mathrm{\ell }\in \mathbb{N}:\text{some path from }u\text{ to }v\text{ has length }\mathrm{\ell }\}\mathrm{.}$

The set being minimized is nonempty by connectivity and finite because a path has no repeated vertex and $V(G)$ is finite (The cardinality $\mid A\mid$ of a finite set). No ordinary natural-number distance is assigned to vertices in different components.

For a vertex $v$, its eccentricity is the maximum of $d_G(v,u)$ over the vertices $u$ in its component. The diameter of a connected graph is the maximum of $d_G(u,v)$ over all pairs of its vertices. Thus every component has its own diameter, while a disconnected graph has no single graph diameter under this convention.

The girth $g(G)$ is the minimum length of a cycle in $G$. If $G$ is acyclic, set $g(G)=\mathrm{\infty }$ by convention. This symbol records absence of a cycle and is not a distance between vertices.