Half-transitive graph

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.

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

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..