Near polygon


In mathematics, a near polygon is an incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in 1980. Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. These structures generalise the notion of generalized polygon as every generalized 2n-gon is a near 2n-gon of a particular kind. Near polygons were extensively studied and connection between them and dual polar spaces was shown in 1980s and early 1990s. Some sporadic simple groups, for example the Hall-Janko group and the Mathieu groups, act as automorphism groups of near polygons.

Definition

A near 2d-gon is an incidence structure, where is the set of points, is the set of lines and is the incidence relation, such that:
Note that the distance are measured in the collinearity graph of points, i.e., the graph formed by taking points as vertices and joining a pair of vertices if they are incident with a common line.
We can also give an alternate graph theoretic definition, a near 2d-gon is a connected graph of finite diameter d with the property that for every vertex x and every maximal clique M there exists a unique vertex x' in M nearest to x.
The maximal cliques of such a graph correspond to the lines in the incidence structure definition.
A near 0-gon is a single point while a near 2-gon is just a single line, i.e., a complete graph. A near quadrangle is same as a generalized quadrangle. In fact, it can be shown that every generalized 2d-gon is a near 2d-gon that satisfies the following two additional conditions:
A near polygon is called dense if every line is incident with at least three points and if every two points at distance two have at least two common neighbours. It is said to have order if every line is incident with precisely s + 1 points and every point is incident with precisely t + 1 lines. Dense near polygons have a rich theory and several classes of them have been completely classified.

Examples

A finite near -gon S is called regular if it has an order and if there exist constants, such that for every two points and at distance, there are precisely lines through containing a point at distance from. It turns out that regular near -gons are precisely those near -gons whose point graph is a distance-regular graph. A generalized -gon of order is a regular near -gon with parameters