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 ≥ 2skew-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-symmetricasymmetric

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..