A generalized hexagon is a generalized polygon
of order 6.

is more commonly known as the Heawood graph, but
is also the -cage graph, the cubic
vertex-transitive graph Ct15, the cubic symmetric
graph ,
the 69-Haar graph, and is an incidence graph of a 2- design.

is the -cage graph, 4137-Haar graph,
and is an incidence graph of a 2- design.

is the Bouwer graph , line graph of the Heawood graph, and is a distance-regular
graph with intersection array ,

is the line graph of the -cage, is also known as the flag graph of (DistanceRegular.org), and is a distance-regular
graph with intersection array .

is the line graph of the -cage, is also known as the flag graph of (DistanceRegular.org), and is the distance-regular
graph with intersection array .

The generalized hexagons are line graphs of the
generalized hexagons .

The following table summarizes some generalized hexagons.

graph | | other names | incidence | graph spectrum |

GH(1, 2) | 14 | Heawood
graph | | |

GH(1,
3) | 26 | (4, 6)-cage
graph, incidence graph of | | |

GH(1, 4) | 42 | (5, 6)-cage
graph | | |

GH(1,
5) | 62 | (6, 6)-cage
graph | | |

GH(1, 7) | 114 | (8, 6)-cage
graph | | |

GH(1, 8) | 146 | (9, 6)-cage
graph | | |

GH(1, 9) | 182 | (10, 6)-cage
graph | | |

GH(2,
1) | 21 | (2,3,7)-Bouwer
graph, flag graph of | | |

GH(2, 8) | 819 | | | |

GH(3, 1) | 52 | | | |

GH(4, 1) | 105 | | | |

GH(5, 1) | 186 | | | |

GH(7, 1) | 456 | | | |

GH(8, 1) | 657 | | | |

GH(8, 2) | 2457 | | | |

## See also

Cage Graph,

Generalized Dodecagon,

Generalized Octagon,

Generalized
Polygon,

Generalized Quadrangle
## Explore with Wolfram|Alpha

## References

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. *Distance-Regular
Graphs.* New York: Springer-Verlag, p. 204, 1989.Brouwer,
A. and Koolen, J. "The Distance-Regular Graphs of Valency Four." *J.
Algebraic Combin.* **10**, 5-24, 1999.DistanceRegular.org. "Flag
Graph of ."
http://www.distanceregular.org/graphs/flag-pg2.3.html.DistanceRegular.org.
"Flag Graph of ." http://www.distanceregular.org/graphs/flag-pg2.4.html.DistanceRegular.org.
"Point Graphs of and its Dual." http://www.distanceregular.org/graphs/point-gh2.2.html.Godsil,
C. and Royle, G. "Two Generalized Hexagons." §5.7 in *Algebraic
Graph Theory.* New York: Springer-Verlag, pp. 88-90, 2001.van
Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their
Spectrum?" *Lin. Algebra Appl.* **373**, 139-162, 2003.## Referenced
on Wolfram|Alpha

Generalized Hexagon
## Cite this as:

Weisstein, Eric W. "Generalized Hexagon."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/GeneralizedHexagon.html

## Subject classifications