Eccentricity of a vertex

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Template:Undirected graph vertex numerical invariant

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