TOPICS
Search

Quartic Symmetric Graph

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
QuarticSymmetricGraphs

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 n=1, 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.

graphnodes
complete graph K_55
octahedral graph (circulant graph Ci_6(1,2))6
circulant graph Ci_8(1,3)8
9-node graph9
cuboctahedral graph12
tesseract16
Doyle graph27
icosidodecahedral graph30

Among quartic symmetric graphs, the 54-node Bouwer graph B(2,6,9) 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.

See also

Cubic Symmetric Graph, Half-Arc-Transitive Graph, Quartic Graph, Quartic Vertex-Transitive Graph, Quintic Symmetric Graph, Symmetric Graph

Explore with Wolfram|Alpha

References

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."

Referenced on Wolfram|Alpha

Quartic Symmetric Graph

Cite this as:

Weisstein, Eric W. "Quartic Symmetric Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuarticSymmetricGraph.html

Subject classifications