edge-transitive graph
{{short description|Graph where all pairs of edges are automorphic}}
{{about|graph theory|edge transitivity in geometry|Edge-transitive}}
{{Graph families defined by their automorphisms}}
In the mathematical field of graph theory, an edge-transitive graph is a graph {{mvar|G}} such that, given any two edges {{math|e{{sub|1}}}} and {{math|e{{sub|2}}}} of {{mvar|G}}, there is an automorphism of {{mvar|G}} that maps {{math|e{{sub|1}}}} to {{math|e{{sub|2}}}}.{{cite book | author=Biggs, Norman | title=Algebraic Graph Theory | edition=2nd | location=Cambridge | publisher=Cambridge University Press | year=1993 | page=118 | isbn=0-521-45897-8}}
In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges.
Examples and properties
Image:Gray graph 2COL.svg is edge-transitive and regular, but not vertex-transitive.]]
The number of connected simple edge-transitive graphs on n vertices is 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, 19, 10, 16, 25, 26, 12, 28 ... {{OEIS|A095424}}
Edge-transitive graphs include all symmetric graphs, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. Every connected edge-transitive graph that is not vertex-transitive must be bipartite, (and hence can be colored with only two colors), and either semi-symmetric or biregular.{{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 = https://books.google.com/books?id=hsymFm0E0uIC&pg=PA20
| year = 2003}}.
Examples of edge but not vertex transitive graphs include the complete bipartite graphs where m ≠ n, which includes the star graphs . For graphs on n vertices, there are (n-1)/2 such graphs for odd n and (n-2) for even n.
Additional edge transitive graphs which are not symmetric can be formed as subgraphs of these complete bi-partite graphs in certain cases. Subgraphs of complete bipartite graphs Km,n exist when m and n share a factor greater than 2. When the greatest common factor is 2, subgraphs exist when 2n/m is even or if m=4 and n is an odd multiple of 6.{{cite journal |last1=Newman |first1=Heather A. |last2=Miranda |first2=Hector |last3=Narayan |first3=Darren A |title=Edge-transitive graphs and combinatorial designs |journal=Involve: A Journal of Mathematics |year=2019 |volume=12 |issue=8 |pages=1329–1341 |doi=10.2140/involve.2019.12.1329 |arxiv=1709.04750 |s2cid=119686233 }} So edge transitive subgraphs exist for K3,6, K4,6 and K5,10 but not K4,10. An alternative construction for some edge transitive graphs is to add vertices to the midpoints of edges of a symmetric graph with v vertices and e edges, creating a bipartite graph with e vertices of order 2, and v of order 2e/v.
An edge-transitive graph that is also regular, but still not vertex-transitive, is called semi-symmetric. The Gray graph, a cubic graph on 54 vertices, is an example of a regular graph which is edge-transitive but not vertex-transitive. The Folkman graph, a quartic graph on 20 vertices is the smallest such graph.
The vertex connectivity of an edge-transitive graph always equals its minimum degree.{{citation
| last = Watkins | first = Mark E.
| doi = 10.1016/S0021-9800(70)80005-9
| journal = Journal of Combinatorial Theory
| mr = 266804
| pages = 23–29
| title = Connectivity of transitive graphs
| volume = 8
| year = 1970| doi-access =
}}
See also
- Edge-transitive (in geometry)
References
{{reflist}}
External links
- {{MathWorld | urlname=Edge-TransitiveGraph | title=Edge-transitive graph }}