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Generalized Dodecagon

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A generalized dodecagon is a generalized polygon of order 12.

GD(1,2) is the (3,12)-cage graph, more commonly known as the Tutte 12-cage.

GD(2,1) is the line graph of the Tutte 12-cage and is a distance-regular graph with intersection array {4,2,2,2,2,2;1,1,1,1,1,2}.

GD(1,3) is a (4,12)-cage graph and incident graph of the generalized hexagon GH(3,3) (DistanceRegular.org). It is also a distance-transitive graph with intersection array {4,3,3,3,3,3;1,1,1,1,1,4}.

GD(1,9) is a distance-transitive graph.

The following table summarizes some generalized dodecagons.

graphVother namesgraph spectrum
GD(1,2)126(-3)^1(-sqrt(6))^(21)(-sqrt(2))^(27)0^(28)(sqrt(2))^(27)(sqrt(6))^(21)3^1
GD(2,1)189flag graph of GH(2,2)(-2)^(64)(1-sqrt(6))^(21)(1-sqrt(2))^(27)1^(28)(1+sqrt(2))^(27)(1+sqrt(6))^(21)4^1
GD(1,3)728(4,12)-cage graph, incidence graph of GH(3,3)(-3)^1(-sqrt(6))^(21)(-sqrt(2))^(27)0^(28)(sqrt(2))^(27)(sqrt(6))^(21)3^1
GD(1,9)132860

See also

Cage Graph, Generalized Hexagon, Generalized Polygon, Generalized Quadrangle, Tutte 12-Cage

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References

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, p. 204, 1989.DistanceRegular.org. "Incidence Graph of GH(3,3)." https://www.math.mun.ca/distanceregular/graphs//incidence-gh3.3.html.van Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their Spectrum?" Lin. Algebra Appl. 373, 139-162, 2003.

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Generalized Dodecagon

Cite this as:

Weisstein, Eric W. "Generalized Dodecagon." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedDodecagon.html

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