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. 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, so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree. The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices.