Diameter of a graph

Definition

The diameter of a graph is defined for any connected graph as the diameter of the metric space induced by it. Explicitly, for a graph $G$ with vertex set $V(G)$, it is:

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

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

Note that for a connected finite graph, the diameter is finite. For an infinite connected graph, the diameter may be finite or $\infty$. A graph of finite diameter is a connected graph (finite or infinite) whose diameter is finite.

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

Related invarants

Invariant	Meaning	Relation with radius
Radius of a graph	minimum of eccentricities of vertices	Radius $\leq$ Diameter $\leq$ Twice the radius

Related graph properties

• Graph of finite diameter
• Graph of constant finite eccentricity