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# Generalized Polygon

Let be an incidence geometry, i.e., a set with a symmetric, reflexive binary relation . Let and be elements of . Let an incidence plane be an incidence geometry whose object set is the disjoint union of two sets and such that for or , only if . Then a generalized polygon is an incidence plane such that for all ,

1. There exists a path of length at most from to , and.

2. There exists at most one irreducible path of length less than from to .

(Feit and Higman 1964).

The only cubic generalized polygons are the generalized 2-gon (utility graph), generalized triangle , generalized quadrangle , and generalized hexagon (Feit and Higman 1964, Royle).

Cage Graph, Generalized Hexagon, Generalized Octagon, Generalized Quadrangle, Moore Graph

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## References

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "Generalized Polygons." §6.5 in Distance Regular Graphs. New York: Springer-Verlag, pp. 200-205, 1989.Feit, W. and Higman, G. "The Non-Existence of Certain Generalized Polygons." J. Algebra 1, 114-131, 1964.Godsil, C. and Royle, G. "Generalized Polygons." §5.6 in Algebraic Graph Theory. New York: Springer-Verlag, pp. 84-87, 2001.Royle, G. "Cubic Cages." http://school.maths.uwa.edu.au/~gordon/remote/cages/.Tits, J. "Sur la trialité et certains groupes qui s'en déduisent." Publ. Math. I.H.E.S. Paris 2, 14-60, 1959.Tits, J. "Théorème de Bruhat er sous-groupes paraboliques." Comptes Rendus Acad. Sci. Paris 254, 2910-2912, 1962.

## Referenced on Wolfram|Alpha

Generalized Polygon

## Cite this as:

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