# Partial geometry

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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 .

## See also

## References

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