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| {{Infobox graph
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| | name = Butterfly graph
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| | image = [[Image:Butterfly graph.svg|200px]]
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| | vertices = 5
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| | edges = 6
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| | automorphisms = 8 ([[Dihedral group|''D'']]<sub>4</sub>)
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| | diameter = 2
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| | radius = 1
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| | girth = 3
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| | chromatic_number = 3
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| | chromatic_index = 4
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| | properties = [[planar graph|Planar]]<br>[[unit distance graph|Unit distance]]<br>[[Eulerian graph|Eulerian]] | |
| }}
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| 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>.
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| 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]].
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| 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>
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| ==Bowtie-free graphs==
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| 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.
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| 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>
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| ==Algebraic properties==
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| 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.
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| The [[characteristic polynomial]] of the butterfly graph is <math>-(x-1)(x+1)^2(x^2-x-4)</math>.
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| == References ==
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| {{reflist}}
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| [[Category:Individual graphs]]
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| [[Category:Planar graphs]]
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