Half-transitive graph

Graph families defined by their automorphisms
distance-transitive	${\displaystyle {\boldsymbol {\rightarrow }}}$	distance-regular	${\displaystyle {\boldsymbol {\leftarrow }}}$	strongly regular
${\displaystyle {\boldsymbol {\downarrow }}}$
symmetric (arc-transitive)	${\displaystyle {\boldsymbol {\leftarrow }}}$	t-transitive, t ≥ 2		skew-symmetric
${\displaystyle {\boldsymbol {\downarrow }}}$
(if connected) vertex- and edge-transitive	${\displaystyle {\boldsymbol {\rightarrow }}}$	edge-transitive and regular	${\displaystyle {\boldsymbol {\rightarrow }}}$	edge-transitive
${\displaystyle {\boldsymbol {\downarrow }}}$		${\displaystyle {\boldsymbol {\downarrow }}}$		${\displaystyle {\boldsymbol {\downarrow }}}$
vertex-transitive	${\displaystyle {\boldsymbol {\rightarrow }}}$	regular	${\displaystyle {\boldsymbol {\rightarrow }}}$	(if bipartite) biregular
${\displaystyle {\boldsymbol {\uparrow }}}$
Cayley graph	${\displaystyle {\boldsymbol {\leftarrow }}}$	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.

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.

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