A quartic symmetric graph is a symmetric graph that is also quartic (i.e., regular of 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 (1970) discovered a class of quartic symmetric graphs, the smallest being the
54-node Bouwer graph, that are not 1-arc-transitive.
An example with 27 nodes (now called the Doyle graph)
was subsequently found by Doyle (1976) and Holt (1981).

Bouwer, Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull.13, 231-237, 1970.Doyle,
P. G. On Transitive Graphs. Senior Thesis. Cambridge, MA, Harvard College,
April 1976.Doyle, P. "A 27-Vertex Graph That Is Vertex-Transitive
and Edge-Transitive But Not L-Transitive." October 1998. http://arxiv.org/abs/math/0703861.Holt,
D. F. "A Graph Which Is Edge Transitive But Not Arc Transitive." J.
Graph Th.5, 201-204, 1981.Sloane, N. J. A. Sequence
A087101 in "The On-Line Encyclopedia
of Integer Sequences."