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# Quartic Symmetric Graph

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.

 graph nodes complete graph 5 octahedral graph (circulant graph ) 6 circulant graph 8 9-node graph 9 cuboctahedral graph 12 tesseract 16 Doyle graph 27 icosidodecahedral graph 30

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).

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

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## References

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."

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Quartic Symmetric Graph

## Cite this as:

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