# Eccentricity of a vertex

## Definition

Suppose  is a connected graph with vertex set  and  is a vertex of . The eccentricity of  is defined as:



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

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 . 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  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