• 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 .
  • (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

Links