# Edge-transitive graph

In the mathematical field of graph theory, an **edge-transitive graph** is a graph *G* such that, given any two edges *e*_{1} and *e*_{2} of *G*, there is an
automorphism of *G* that maps *e*_{1} to *e*_{2}.[1]

Graph families defined by their automorphisms | ||||
---|---|---|---|---|

distance-transitive | → | distance-regular | ← | strongly regular |

↓ | ||||

symmetric (arc-transitive) | ← | t-transitive, t ≥ 2 |
skew-symmetric | |

↓ | ||||

_{(if connected)}vertex- and edge-transitive |
→ | edge-transitive and regular | → | edge-transitive |

↓ | ↓ | ↓ | ||

vertex-transitive | → | regular | → | _{(if bipartite)}biregular |

↑ | ||||

Cayley graph | ← | zero-symmetric | asymmetric |

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 that is not vertex-transitive must be bipartite and either semi-symmetric or biregular.[2]

The vertex connectivity of an edge-transitive graph always equals its minimum degree.[3]

## See also

- Edge-transitive (in geometry)

## References

- Biggs, Norman (1993).
*Algebraic Graph Theory*(2nd ed.). Cambridge: Cambridge University Press. p. 118. ISBN 0-521-45897-8. - Lauri, Josef; Scapellato, Raffaele (2003),
*Topics in Graph Automorphisms and Reconstruction*, London Mathematical Society Student Texts, Cambridge University Press, pp. 20–21, ISBN 9780521529037. - Watkins, Mark E. (1970), "Connectivity of transitive graphs",
*Journal of Combinatorial Theory*,**8**: 23–29, doi:10.1016/S0021-9800(70)80005-9, MR 0266804