Barycentric subdivision: Difference between revisions
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{{Graph families defined by their automorphisms}} | |||
{{about|graph theory|edge transitivity in geometry|Edge-transitive}} | |||
In the [[mathematics|mathematical]] field of [[graph theory]], an '''edge-transitive graph''' is a [[Graph (mathematics)|graph]] ''G'' such that, given any two edges ''e''<sub>1</sub> and ''e''<sub>2</sub> of ''G'', there is an | |||
[[Graph automorphism|automorphism]] of ''G'' that maps ''e''<sub>1</sub> to ''e''<sub>2</sub>.<ref name="biggs">{{cite book | author=Biggs, Norman | title=Algebraic Graph Theory | edition=2nd ed. | location=Cambridge | publisher=Cambridge University Press | year=1993 | page=118 | isbn=0-521-45897-8}}</ref> | |||
In other words, a graph is edge-transitive if its [[automorphism group]] acts [[group action|transitively]] upon its edges. | |||
==Examples and properties== | |||
[[Image:Gray graph 2COL.svg|thumb|200px|The [[Gray graph]] is edge-transitive and [[regular graph|regular]], but not [[vertex-transitive graph|vertex-transitive]].]] | |||
Edge-transitive graphs include any [[complete bipartite graph]] <math>K_{m,n}</math>, and any [[symmetric graph]], such as the vertices and edges of the [[cube]].<ref name="biggs" /> Symmetric graphs are also [[vertex-transitive graph|vertex-transitive]] (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. The [[Gray graph]] is an example of a graph which is edge-transitive but not vertex-transitive. All such graphs are [[bipartite graph|bipartite]],<ref name="biggs" /> and hence can be [[Graph coloring|colored]] with only two colors. | |||
An edge-transitive graph that is also [[regular graph|regular]], but not vertex-transitive, is called [[semi-symmetric graph|semi-symmetric]]. The [[Gray graph]] again provides an example. | |||
Every edge-transitive graph must be [[bipartite graph|bipartite]] and either semi-symmetric or [[biregular graph|biregular]].<ref name="ls03">{{citation | |||
| last1 = Lauri | first1 = Josef | |||
| last2 = Scapellato | first2 = Raffaele | |||
| isbn = 9780521529037 | |||
| pages = 20–21 | |||
| publisher = Cambridge University Press | |||
| series = London Mathematical Society Student Texts | |||
| title = Topics in Graph Automorphisms and Reconstruction | |||
| url = http://books.google.com/books?id=hsymFm0E0uIC&pg=PA20 | |||
| year = 2003}}.</ref> | |||
== See also == | |||
* [[Edge-transitive]] (in geometry) | |||
==References== | |||
{{reflist}} | |||
== External links == | |||
* {{MathWorld | urlname=Edge-TransitiveGraph | title=Edge-transitive graph }} | |||
[[Category:Graph families]] | |||
[[Category:Algebraic graph theory]] | |||
Latest revision as of 14:59, 19 September 2013
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In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an
automorphism of G that maps e1 to e2.[1]
In other words, a graph is edge-transitive if its automorphism group acts transitively upon its edges.
Examples and properties
Edge-transitive graphs include any complete bipartite graph , and any symmetric graph, such as the vertices and edges of the cube.[1] Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive. All such graphs are bipartite,[1] and hence can be colored with only two colors.
An edge-transitive graph that is also regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example. Every edge-transitive graph must be bipartite and either semi-symmetric or biregular.[2]
See also
- Edge-transitive (in geometry)
References
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External links
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