Radius of a graph - Revision history

Vipul at 20:02, 27 May 2012

2012-05-27T20:02:08Z

← Older revision		Revision as of 20:02, 27 May 2012
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	==Definition==		==Definition==

	The '''radius of a graph''' is defined for any [[connected graph]] as the radius of the [[metric space induced by a connected ~~space~~\|metric space induced by it]]. Explicitly, for a graph <math>G</matH> with vertex set <math>V(G)</math>, it is:		The '''radius of a graph''' is defined for any [[connected graph]] as the radius of the [[metric space induced by a connected graph\|metric space induced by it]]. Explicitly, for a graph <math>G</matH> with vertex set <math>V(G)</math>, it is:

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

Vipul at 19:59, 27 May 2012

2012-05-27T19:59:44Z

← Older revision		Revision as of 19:59, 27 May 2012
Line 3:		Line 3:
	==Definition==		==Definition==

	The '''radius of a graph''' is defined for any [[connected graph]] as the radius of the [[metric space induced by a connected space\|metric space induced by it]]. Explicitly, it is:		The '''radius of a graph''' is defined for any [[connected graph]] as the radius of the [[metric space induced by a connected space\|metric space induced by it]]. Explicitly, for a graph <math>G</matH> with vertex set <math>V(G)</math>, it is:

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

	where <math>d</math> denotes the distance between two vertices. In words, the radius of a graph is the minimum, over all vertices, of the [[eccentricity of a vertex\|eccentricity of that vertex]].		where <math>d</math> denotes the distance between two vertices. In words, the radius of a graph is the minimum, over all vertices, of the [[eccentricity of a vertex\|eccentricity of that vertex]].

Vipul at 19:58, 27 May 2012

2012-05-27T19:58:16Z

← Older revision		Revision as of 19:58, 27 May 2012
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	<math>\min_{x \in G} \left(\max_{y \in G} d(x,y)\right)</math>		<math>\min_{x \in G} \left(\max_{y \in G} d(x,y)\right)</math>

	where <math>d</math> denotes the distance between two vertices.		where <math>d</math> denotes the distance between two vertices. In words, the radius of a graph is the minimum, over all vertices, of the [[eccentricity of a vertex\|eccentricity of that vertex]].

	Note that for a [[finite graph]], the radius is finite. For an infinite graph, the radius may be finite or <math>\infty</math>.		Note that for a [[finite graph]], the radius is finite. For an infinite graph, the radius may be finite or <math>\infty</math>.
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	For a graph that is not connected, we can consider the radius to be either <math>\infty</math> or undefined.		For a graph that is not connected, we can consider the radius to be either <math>\infty</math> or undefined.

	==Related ~~notions~~==		==Related invarants==

	* [[Diameter of a graph]]		{\| class="sortable" border="1"
	* [[Eccentricity of a graph]]		! Invariant !! Meaning !! Relation with radius
			\|-
			\| [[Diameter of a graph]] \|\| maximum of distances between pairs of vertices \|\| Radius <math>\le</math> Diameter <math>\le</math> Twice the radius
			\|}

Vipul: Created page with "{{undirected graph numerical invariant}} ==Definition== The '''radius of a graph''' is defined for any connected graph as the radius of the [[metric space induced by a c..."

2012-05-27T19:55:36Z

Created page with "{{undirected graph numerical invariant}} ==Definition== The '''radius of a graph''' is defined for any connected graph as the radius of the [[metric space induced by a c..."

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{{undirected graph numerical invariant}}

==Definition==

The '''radius of a graph''' is defined for any [[connected graph]] as the radius of the [[metric space induced by a connected space|metric space induced by it]]. Explicitly, it is:

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

where <math>d</math> denotes the distance between two vertices.

Note that for a [[finite graph]], the radius is finite. For an infinite graph, the radius may be finite or <math>\infty</math>.

For a graph that is not connected, we can consider the radius to be either <math>\infty</math> or undefined.

==Related notions==

* [[Diameter of a graph]]
* [[Eccentricity of a graph]]