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| 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 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.
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. Archived from the original on 2010-06-11. Retrieved 2009-09-05.
- ^ Bouwer, Z. (1970). "Vertex and Edge Transitive, But Not 1-Transitive Graphs". Canadian Mathematical Bulletin. 13: 231–237. doi:10.4153/CMB-1970-047-8.
- ^ 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..