In graph theory, the Holt graph or Doyle graph is the smallest half-transitive graph, that is, the smallest example of a vertex-transitive and edge-transitive graph which is not also symmetric.Doyle, P. "A 27-Vertex Graph That Is Vertex-Transitive and Edge-Transitive But Not L-Transitive." October 1998. . Such graphs are not common.Jonathan L. Gross, Jay Yellen, Handbook of Graph Theory, CRC Press, 2004, , p. 491. It is named after Peter G. Doyle and Derek F. Holt, who discovered the same graph independently in 1976. As cited by MathWorld. and 1981. respectively.
The Holt graph has diameter 3, radius 3 and girth 5, chromatic number 3, chromatic index 5 and is Hamiltonian with distinct Hamiltonian cycles. It is also a 4-vertex-connected and a 4-edge-connected graph. It has book thickness 3 and queue number 3.Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018 The graph is not 1-planar..
It has an automorphism group of order 54. This is a smaller group than a symmetric graph with the same number of vertices and edges would have. The graph drawing on the right highlights this, in that it lacks reflectional symmetry.
The characteristic polynomial of the Holt graph is
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