View source for Dipole graph

{{Short description|Multigraph with two vertices}}
{{No footnotes|date=November 2024}}
{{infobox graph
 | name = Dipole graph
 | image = [[Image:Dipole graph.svg|140px]]
 | image_caption = 
 | vertices = 2
 | edges = {{mvar|n}}
 | chromatic_number = 2 (for {{math|''n'' ≥ 1}})
 | chromatic_index = {{mvar|n}}
 | diameter = 1 (for {{math|''n'' ≥ 1}})
 | properties = [[Connected graph|connected]] (for {{math|''n'' ≥ 1}})<br>[[planar graph|planar]]
}}

In [[graph theory]], a '''dipole graph''', '''dipole''', '''bond graph''', or '''linkage''', is a [[multigraph]] consisting of two [[vertex (graph theory)|vertices]] connected with a number of [[Multiple edges|parallel edges]]. A dipole graph containing {{mvar|n}} edges is called the {{nowrap|size-{{mvar|n}}}} dipole graph, and is denoted by {{math|D{{sub|''n''}}}}. The {{nowrap|size-{{mvar|n}}}} dipole graph is [[dual graph|dual]] to the [[cycle graph]] {{math|C{{sub|''n''}}}}.

The [[Hexagonal lattice|honeycomb]] as an abstract graph is the maximal abelian [[covering graph]] of the dipole graph {{math|D{{sub|3}}}}, while the [[Diamond cubic|diamond crystal]] as an abstract graph is the maximal abelian covering graph of {{math|D{{sub|4}}}}.

Similarly to the [[Platonic graph]]s, the dipole graphs form the skeletons of the [[hosohedron|hosohedra]]. Their duals, the cycle graphs, form the skeletons of the [[dihedron|dihedra]].

==References==

* {{MathWorld|title = Dipole Graph|urlname = DipoleGraph}}
* Jonathan L. Gross and Jay Yellen, 2006. ''Graph Theory and Its Applications, 2nd Ed.'', p.&nbsp;17. Chapman & Hall/CRC. {{ISBN|1-58488-505-X}}
* [[Toshikazu Sunada|Sunada T.]], ''Topological Crystallography, With a View Towards Discrete Geometric Analysis'',  Springer, 2013, {{ISBN|978-4-431-54176-9}} (Print) 978-4-431-54177-6 (Online)

[[Category:Extensions and generalizations of graphs]]
[[Category:Parametric families of graphs]]
[[Category:Regular graphs]]


{{graph-stub}}