Vipul: Created page with "{{undirected graph vertex numerical invariant}} ==Definition== Suppose $G$ is a connected graph with vertex set $V(G)$ and $x$ is a vert..."

2012-05-27T20:05:41Z

Created page with "{{undirected graph vertex numerical invariant}} ==Definition== Suppose <math>G</matH> is a connected graph with vertex set <math>V(G)</math> and <math>x</math> is a vert..."

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{{undirected graph vertex numerical invariant}}

==Definition==

Suppose <math>G</matH> is a [[connected graph]] with vertex set <math>V(G)</math> and <math>x</math> is a vertex of <math>G</math>. The '''eccentricity''' of <math>x</math> is defined as:

<math>\max_{y \in V(G)} d(x,y)</math>

where <math>d</math> denotes the distance between two vertices in the [[metric space induced by a connected graph|metric space induced by the connected graph]] <math>G</math>.

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 <math>\infty</math>. 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 <math>\infty</math> 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==

{| class="sortable" border="1"
! 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
|}

Eccentricity of a vertex - Revision history

Vipul: Created page with "{{undirected graph vertex numerical invariant}} ==Definition== Suppose G is a connected graph with vertex set V(G) and x is a vert..."

Vipul: Created page with "{{undirected graph vertex numerical invariant}} ==Definition== Suppose $G$ is a connected graph with vertex set $V(G)$ and $x$ is a vert..."