Radius of a graph

Definition

The radius of a graph is defined for any connected graph as the radius of the metric space induced by it. Explicitly, it is:

$\min _{x\in G}\left(\max _{y\in G}d(x,y)\right)$

where $d$ denotes the distance between two vertices. In words, the radius of a graph is the minimum, over all vertices, of the eccentricity of that vertex.

Note that for a finite graph, the radius is finite. For an infinite graph, the radius may be finite or $\infty$.

For a graph that is not connected, we can consider the radius to be either $\infty$ or undefined.

Related invarants

Invariant	Meaning	Relation with radius
Diameter of a graph	maximum of distances between pairs of vertices	Radius $\leq$ Diameter $\leq$ Twice the radius