A quartic symmetric graph is a regular graph that is both vertex-transitive and edge-transitive
and has vertex degree 4. The numbers of symmetric quartic graphs on , 2, ... are 0, 0, 0, 0, 1, 1, 0, 1, 1, ... (OEIS A087101).
Some quartic symmetric graphs are illustrated above and listed in the following table.
Among quartic symmetric graphs, the 54-node Bouwer graph constructed by Bouwer (1970)
is half-arc-transitive. The smaller
27-node Doyle graph was independently discovered by
Doyle (1976) and Holt (1981).
Among quartic symmetric graphs, Potočnik et al. (2013) enumerated the arc-transitive subclass through 640 vertices, comprising 4,820 connected
graphs.
Bouwer, I. Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull.13, 231-237, 1970.
https://doi.org/10.4153/CMB-1970-047-8.Doyle,
P. "A 27-Vertex Graph That Is Vertex-Transitive and Edge-Transitive But Not
L-Transitive." October 1998. https://arxiv.org/abs/math/0703861.Doyle,
P. G. On Transitive Graphs. Senior Thesis. Cambridge, MA, Harvard College,
April 1976.Holt, D. F. "A Graph Which Is Edge Transitive But
Not Arc Transitive." J. Graph Th.5, 201-204, 1981.Potočnik,
P.; Spiga, P.; and Verret, G. "Cubic Vertex-Transitive Graphs on Up to 1280
Vertices." J. Symb. Comput.50, 465-477, 2013. https://doi.org/10.1016/j.jsc.2012.09.002.Sloane,
N. J. A. Sequence A087101 in "The
On-Line Encyclopedia of Integer Sequences."
, 2, ... are 0, 0, 0, 0, 1, 1, 0, 1, 1, ... (OEIS A087101).
Some quartic symmetric graphs are illustrated above and listed in the following table.
)
constructed by Bouwer (1970)
is half-arc-transitive. The smaller
27-node Doyle graph was independently discovered by
Doyle (1976) and Holt (1981).
