TOPICS

# Generalized Hexagon

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

Cage Graph, Generalized Dodecagon, Generalized Octagon, Generalized Polygon, Generalized Quadrangle

## Explore with Wolfram|Alpha

More things to try:

## 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