Half-transitive graph

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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

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  3. Bouwer, Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull. 13, 231–237, 1970.
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