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 .
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. Algebra1, 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. Paris2, 14-60, 1959.Tits, J. "Théorème
de Bruhat er sous-groupes paraboliques." Comptes Rendus Acad. Sci. Paris254,
2910-2912, 1962.
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 
,
from 
to 
, and.
from 
to 
.
(utility graph), generalized triangle 
, generalized quadrangle 
, and generalized
hexagon 
(Feit and Higman 1964, Royle).
