Partial geometry
From Infogalactic: the planetary knowledge core
An incidence structure consists of points , lines , and flags where a point is said to be incident with a line if . It is a (finite) partial geometry if there are integers such that:
- For any pair of distinct points and , there is at most one line incident with both of them.
- Each line is incident with points.
- Each point is incident with lines.
- If a point and a line are not incident, there are exactly pairs , such that is incident with and is incident with .
A partial geometry with these parameters is denoted by .
Properties
- The number of points is given by and the number of lines by .
- The point graph of a is a strongly regular graph : .
- Partial geometries are dual structures : the dual of a is simply a .
Special case
- The generalized quadrangles are exactly those partial geometries with .
- The Steiner systems are precisely those partial geometries with .
Generalisations
A partial linear space of order is called a semipartial geometry if there are integers such that:
- If a point and a line are not incident, there are either or exactly pairs , such that is incident with and is incident with .
- Every pair of non-collinear points have exactly common neighbours.
A semipartial geometry is a partial geometry if and only if .
It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters .
A nice example of such a geometry is obtained by taking the affine points of and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters .
See also
References
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