# 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. and 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..