In combinatorial theory, a generalized polygon is an incidence structure introduced by Jacques Tits. Generalized polygons encompass as special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss. Every generalized polygon is also a near polygon.
A generalized 2-gon (or a digon) is a partial linear space where each point is incident to each line. For n > 3 a generalized n-gon is an incidence structure (), where is the set of points, is the set of lines and is the incidence relation, such that:
- It is a partial linear space.
- It has no ordinary m-gons as subgeometry for 2 < m < n.
- It has ordinary n-gon as a subgeometry.
- For any there exists a subgeometry () isomorphic to an ordinary n-gon such that .
An equivalent but sometimes simpler way to express these conditions is: consider the bipartite incidence graph with the vertex set and the edges connecting the incident pairs of points and lines.
A generalized polygon is of order (s,t) if:
- all vertices of the incidence graph corresponding to the elements of have the same degree s + 1 for some natural number s; in other words, every line contains exactly s + 1 points,
- all vertices of the incidence graph corresponding to the elements of have the same degree t + 1 for some natural number t; in other words, every point lies on exactly t + 1 lines.
We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order.
The dual of a generalized n-gon (), is the incidence structure with notion of points and lines reversed and the incidence relation taken to be the inverse relation of . It can easily be shown that this is again a generalized n-gon.
- A generalized digon is a complete bipartite graph Ks+1,t+1.
- For any natural n ≥ 3, consider the boundary of the ordinary polygon with n sides. Declare the vertices of the polygon to be the points and the sides to be the lines, with set inclusion as the incidence relation. This results in a generalized n-gon with s = t = 1.
- For each group of Lie type G of rank 2 there is an associated generalized n-gon X with n equal to 3, 4, 6 or 8 such that G acts transitively on the set of flags of X. In the finite case, for n=6, one obtains the Split Cayley hexagon of order (q,q) for G2(q) and the twisted triality hexagon of order (q3,q) for 3D4(q3), and for n=8, one obtains the Ree-Tits octagon of order (q,q2) for 2F4(q) with q=22n+1. Up to duality, these are the only known thick finite generalized hexagons or octagons.
- 2, 3, 4, 6 or 8.
- If n = 2, the structure is a complete bipartite graph.
- If n = 3, the structure is a finite projective plane, and s = t.
- If n = 4, the structure is a finite generalized quadrangle, and t1/2 ≤ s ≤ t2.
- If n = 6, then st is a square, and t1/3 ≤ s ≤ t3.
- If n = 8, then 2st is a square, and t1/2 ≤ s ≤ t2.
- If s or t is allowed to be 1 and the structure is not the ordinary n-gon then besides the values of n already listed, only n = 12 may be possible.
If s and t are both infinite then generalized polygons exist for each n greater or equal to 2. It is unknown whether or not there exist generalized polygons with one of the parameters finite and the other infinite (these cases are called semi-finite).
The incidence graphs of generalized polygons have important properties. For example, every generalized n-gon of order (s,s) is a (s,2n) cage. They are also related to expander graphs as they have nice expansion properties. Several classes of extremal expander graphs are obtained from generalized polygons.
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