Eccentricity of a vertex

Definition

Suppose $G$ is a connected graph with vertex set $V(G)$ and $x$ is a vertex of $G$. The eccentricity of $x$ is defined as:

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

where $d$ denotes the distance between two vertices in the metric space induced by the connected graph $G$.

Note that for a connected finite graph, the eccentricity of every vertex is finite. For an infinite connected graph, the eccentricity of a vertex may be finite or $\infty$. However, it remains true that either all eccentricities are finite or all eccentricities are infinite.

For a graph that is not connected, all eccentricities can be considered to be $\infty$ or undefined.

Facts

• Given any two vertices of a connected graph, the eccentricity of either vertex is at most twice the eccentricity of the other vertex.

Related invarants

Invariant	Meaning in terms of eccentricity
Radius of a graph	minimum of eccentricities of all vertices
Diameter of a graph	maximum of eccentricities of all vertices