= Graph radius
{title2=$\operatorname{rad}G=\min_v e_G(v)$}
The <graph radius> of a finite <connected graph> $G$ is the least <graph eccentricity> of any vertex:
$$
\operatorname{rad}G=\min_{v\in V(G)}\max_{u\in V(G)}d_G(v,u).
$$
It is the smallest worst-case travel distance achievable by choosing one starting vertex. The vertices achieving it form the <center of a graph>.
For the five-vertex <path graph>, the eccentricities are $4,3,2,3,4$, so the radius is two and the <graph diameter> is four. Every finite <connected graph> satisfies $\operatorname{rad}G\le\operatorname{diam}G\le2\operatorname{rad}G$: choose a central vertex $c$ and apply the <triangle inequality> $d(u,v)\le d(u,c)+d(c,v)$. In a five-cycle every vertex has eccentricity two, so radius and diameter coincide. A vertex minimizing the sum of distances solves a different optimization problem.
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