Half-transitive graph

Graph families defined by their automorphisms
distance-transitive	${\boldsymbol {\rightarrow }}$	distance-regular	${\boldsymbol {\leftarrow }}$	strongly regular
${\boldsymbol {\downarrow }}$
symmetric (arc-transitive)	${\boldsymbol {\leftarrow }}$	t-transitive, t ≥ 2		skew-symmetric
${\boldsymbol {\downarrow }}$
(if connected) vertex- and edge-transitive	${\boldsymbol {\rightarrow }}$	edge-transitive and regular	${\boldsymbol {\rightarrow }}$	edge-transitive
${\boldsymbol {\downarrow }}$		${\boldsymbol {\downarrow }}$		${\boldsymbol {\downarrow }}$
vertex-transitive	${\boldsymbol {\rightarrow }}$	regular	${\boldsymbol {\rightarrow }}$	(if bipartite) biregular
${\boldsymbol {\uparrow }}$
Cayley graph	${\boldsymbol {\leftarrow }}$	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. Handbook of Combinatorics. Elsevier.
3. Bouwer, Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull. 13, 231–237, 1970.
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..