-
A generalized quadrangle is a partial linear space where
- Given any line
and point not on . There is a unique point on such that and are collinear. - There are noncollinear points and nonconcurrent lines.
- (Godsil 5.4.1) Let
be a partial linear space that contains noncollinear points and nonconcurrent lines. Then is a generalized quadrangle if and only if its incidence graph has diameter and girth .
- Given any line
-
(Godsil 5.5.1) Let
be the incidence structure where the points and lines are the totally isotropic points and totally isotropic lines of . Then is a generalized quadrangle -
A generalized polygon is a finite bipartite graph with diameter
and girth . We also refer to this as a generalized -gon. -
A vertex
in a generalized polygon is thick if . Otherwise, they are thin. A generalized polygon is thick if all its vertices are thick.
-
Let
be a generalized -gon. -
(Godsil 5.6.1) If
, then there is a unique path of length from to . -
(Godsil 5.6.2)
. -
(Godsil 5.6.3) Every vertex in
has degree at least two. -
(Godsil 5.6.4) Any two vertices lie in a cycle of length
. -
(Godsil 5.6.5) Let
be a cycle of length , any two vertices at the same distance in from a thick vertex in have the same degree. -
(Godsil 5.6.6) The minimum distance
between any pair of thick vertices is a divisor of . If
is odd, then all thick vertices have the same degree. If
is even, then the thick vertices have at most two degrees. Any vertices at distance
from a thick vertex is itself thick -
(Godsil 5.6.7) If
is not thick, it is either a cycle, a -fold subdivision of a multiple edge, or the -fold subdivision of a thick generalized polygon.
-
-
(Godsil 5.6.8) Feit-Higman Theorem If a generalized
-gon is thick, then . -
(Godsil 5.6.9) If a generalized polygon is regular it is distance regular.
-
If the degrees of the vertices of a thick generalized polygon are
, we say that it has order . -
(Godsil 5.6.10) Let
be a thick generalized -gon of order . WLOG suppose - If
, then - If
, then for some . and - If
, then for some and
- If