# Half-transitive graph

In the mathematical field of graph theory, a **half-transitive graph** is a graph that is both vertex-transitive and edge-transitive, but not symmetric.[1] In other words, a graph is half-transitive if its automorphism group acts transitively upon both its vertices and its edges, but not on ordered pairs of linked vertices.

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 |

Every connected symmetric graph must be vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree,[2] so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree.[3] The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices.[4][5]

## References

- Gross, J.L.; Yellen, J. (2004).
*Handbook of Graph Theory*. CRC Press. p. 491. ISBN 1-58488-090-2. - Babai, L (1996). "Automorphism groups, isomorphism, reconstruction". In Graham, R; Grötschel, M; Lovász, L (eds.).
*Handbook of Combinatorics*. Elsevier. - Bouwer, Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull. 13, 231–237, 1970.
- Biggs, Norman (1993).
*Algebraic Graph Theory*(2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-45897-8. - Holt, Derek F. (1981). "A graph which is edge transitive but not arc transitive".
*Journal of Graph Theory*.**5**(2): 201–204. doi:10.1002/jgt.3190050210..

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