Generalized Polygon


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

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

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

(Feit and Higman 1964).

The only cubic generalized polygons are the generalized 2-gon K_(3,3) (utility graph), generalized triangle PG(2,2), generalized quadrangle W_2, and generalized hexagon GH(2,2) (Feit and Higman 1964, Royle).

See also

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

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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.", 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.

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