Overconstrained mechanism: Difference between revisions
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{{Infobox graph | |||
| name = Butterfly graph | |||
| image = [[Image:Butterfly graph.svg|200px]] | |||
| vertices = 5 | |||
| edges = 6 | |||
| automorphisms = 8 ([[Dihedral group|''D'']]<sub>4</sub>) | |||
| diameter = 2 | |||
| radius = 1 | |||
| girth = 3 | |||
| chromatic_number = 3 | |||
| chromatic_index = 4 | |||
| properties = [[planar graph|Planar]]<br>[[unit distance graph|Unit distance]]<br>[[Eulerian graph|Eulerian]] | |||
}} | |||
In the [[mathematics|mathematical]] field of [[graph theory]], the '''butterfly graph''' (also called the '''bowtie graph''' and the '''hourglass graph''') is a [[planar graph|planar]] [[undirected graph]] with 5 vertices and 6 edges.<ref>{{MathWorld|urlname=ButterflyGraph|title=Butterfly Graph}}</ref><ref>ISGCI: Information System on Graph Classes and their Inclusions. "[http://www.graphclasses.org/smallgraphs.html List of Small Graphs]".</ref> It can be constructed by joining 2 copies of the [[cycle graph]] ''C''<sub>3</sub> with a common vertex and is therefore isomorphic to the [[friendship graph]] ''F''<sub>2</sub>. | |||
The butterfly Graph has [[graph diameter|diameter]] 2 and [[girth (graph theory)|girth]] 3, radius 1, [[chromatic number]] 3, [[chromatic index]] 4 and is both [[Eulerian graph|Eulerian]] and [[unit distance graph|unit distance]]. It is also a 1-[[k-vertex-connected graph|vertex-connected graph]] and a 2-[[k-edge-connected graph|edge-connected graph]]. | |||
There are only 3 [[Graceful labeling|non-graceful]] simple graphs with five vertices. One of them is the butterfly graph. The two others are [[cycle graph]] ''C''<sub>5</sub> and the [[complete graph]] ''K''<sub>5</sub>.<ref name="Mat2007">{{mathworld|title=Graceful graph|urlname=GracefulGraph}}</ref> | |||
==Bowtie-free graphs== | |||
A graph is '''bowtie-free''' if it has no butterfly as an [[induced subgraph]]. The [[triangle-free graph]]s are bowtie-free graphs, since every butterfly contains a triangle. | |||
In a [[k-vertex-connected graph|''k''-vertex-connected]] graph, and edge is said ''k''-contractible if the contraction of the edge results in a ''k''-connected graph. Ando, Kaneko, Kawarabayashi and Yoshimoto proved that every ''k''-vertex-connected bowtie-free graph has a ''k''-contractible edge.<ref>Kiyoshi Ando "Contractible Edges in a k-Connected Graph", CJCDGCGT 2005: 10-20 [http://www.combinatorics.net/conf/conf/abstract/ando.htm].</ref> | |||
==Algebraic properties== | |||
The full automorphism group of the butterfly graph is a group of order 8 isomorphic to the [[Dihedral group]] ''D''<sub>4</sub>, the group of symmetries of a [[Square (geometry)|square]], including both rotations and reflections. | |||
The [[characteristic polynomial]] of the butterfly graph is <math>-(x-1)(x+1)^2(x^2-x-4)</math>. | |||
== References == | |||
{{reflist}} | |||
[[Category:Individual graphs]] | |||
[[Category:Planar graphs]] | |||
Revision as of 02:21, 31 December 2013
In the mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar undirected graph with 5 vertices and 6 edges.[1][2] It can be constructed by joining 2 copies of the cycle graph C3 with a common vertex and is therefore isomorphic to the friendship graph F2.
The butterfly Graph has diameter 2 and girth 3, radius 1, chromatic number 3, chromatic index 4 and is both Eulerian and unit distance. It is also a 1-vertex-connected graph and a 2-edge-connected graph.
There are only 3 non-graceful simple graphs with five vertices. One of them is the butterfly graph. The two others are cycle graph C5 and the complete graph K5.[3]
Bowtie-free graphs
A graph is bowtie-free if it has no butterfly as an induced subgraph. The triangle-free graphs are bowtie-free graphs, since every butterfly contains a triangle.
In a k-vertex-connected graph, and edge is said k-contractible if the contraction of the edge results in a k-connected graph. Ando, Kaneko, Kawarabayashi and Yoshimoto proved that every k-vertex-connected bowtie-free graph has a k-contractible edge.[4]
Algebraic properties
The full automorphism group of the butterfly graph is a group of order 8 isomorphic to the Dihedral group D4, the group of symmetries of a square, including both rotations and reflections.
The characteristic polynomial of the butterfly graph is .
References
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- ↑
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