Central vertex: Difference between revisions
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==Definition== | ==Definition== | ||
Suppose <math>G</math> is a [[connected graph|connected]] [[undirected graph]] that has a finite [[defining ingredient::radius of a graph|radius]]. (Note that any finite connected undirected graph must have finite radius, but infinite graphs may sometimes have finite radius). A vertex <math>v</math> of <math>G</math> is termed a '''central vertex''' if the [[defining ingredient::eccentricity of a vertex|eccentricity]] of <math>v</math> (i.e., the maximum of distances between <math>v</math> and vertices of <math>G</math>) equals the radius of <math>G</math>. Note that the radius is defined as the minimum of the eccentricities of all vertices. | Suppose <math>G</math> is a [[graph of finite diameter]], i.e., it is a [[connected graph|connected]] [[undirected graph]] that has a finite [[defining ingredient::radius of a graph|radius]]. (Note that any finite connected undirected graph must have finite radius, but infinite graphs may sometimes have finite radius). A vertex <math>v</math> of <math>G</math> is termed a '''central vertex''' if the [[defining ingredient::eccentricity of a vertex|eccentricity]] of <math>v</math> (i.e., the maximum of distances between <math>v</math> and vertices of <math>G</math>) equals the radius of <math>G</math>. Note that the radius is defined as the minimum of the eccentricities of all vertices. | ||
==Facts== | ==Facts== | ||
Latest revision as of 17:32, 28 May 2012
Template:Undirected graph vertex property
Definition
Suppose is a graph of finite diameter, i.e., it is a connected undirected graph that has a finite radius. (Note that any finite connected undirected graph must have finite radius, but infinite graphs may sometimes have finite radius). A vertex of is termed a central vertex if the eccentricity of (i.e., the maximum of distances between and vertices of ) equals the radius of . Note that the radius is defined as the minimum of the eccentricities of all vertices.
Facts
- Tree has either one central vertex or two adjacent central vertices
- Connected graph may have precisely two non-adjacent central vertices
Related properties
- Peripheral vertex is in some sense the opposite: it is a vertex whose eccentriciti equals the diameter, which is the maximum of all eccentricities.