Edge-transitive graph

This article is about graph theory. For edge transitivity in geometry, see Edge-transitive.
Graph families defined by their automorphisms
distance-transitive\boldsymbol{\rightarrow}distance-regular\boldsymbol{\leftarrow}strongly regular
\boldsymbol{\downarrow}
symmetric (arc-transitive)\boldsymbol{\leftarrow}t-transitive, t ≥ 2skew-symmetric
\boldsymbol{\downarrow}
(if connected)
vertex- and edge-transitive		\boldsymbol{\rightarrow}edge-transitive and regular\boldsymbol{\rightarrow}edge-transitive
\boldsymbol{\downarrow}\boldsymbol{\downarrow}\boldsymbol{\downarrow}
vertex-transitive\boldsymbol{\rightarrow}regular\boldsymbol{\rightarrow}(if bipartite)
biregular
\boldsymbol{\uparrow}
Cayley graph\boldsymbol{\leftarrow}zero-symmetricasymmetric

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

The Gray graph is edge-transitive and regular, but not vertex-transitive.

Edge-transitive graphs include any complete bipartite graph K_{m,n}, 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 that is not vertex-transitive must be bipartite and either semi-symmetric or biregular.[2]

See also

References

  1. 1 2 3 Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge: Cambridge University Press. p. 118. ISBN 0-521-45897-8.
  2. ↑ Lauri, Josef; Scapellato, Raffaele (2003), Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts, Cambridge University Press, pp. 20–21, ISBN 9780521529037.

External links

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