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.
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-vertex Doyle
graph was independently discovered by Doyle (1976) and Holt (1981).
