Article · Wikipedia archive · Last revised Jun 2, 2026

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. 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.

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Jun 2, 2026
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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.

The Holt graph is the smallest half-transitive graph. The lack of reflectional symmetry in this drawing highlights the fact that edges are not equivalent to their inverse. source ↗

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.45

References

References

  1. Gross, J.L.; Yellen, J. (2004). Handbook of Graph Theory. CRC Press. p. 491. ISBN 1-58488-090-2.
  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.
  3. Bouwer, Z. (1970). "Vertex and Edge Transitive, But Not 1-Transitive Graphs". Canadian Mathematical Bulletin. 13 (2): 231–237. doi:10.4153/CMB-1970-047-8.
  4. Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-45897-8.
  5. 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..