An incidence structure consists of a set of points, a set is said to be incident if . It is a (finite) partial geometry if there are integers such that:

• and , there is at most one line incident with both of them.

• points.

• lines.

• pairs is incident with .

A partial geometry with these parameters is denoted by .

Properties

• and the number of lines by .

• The point graph (also known as the collinearity graph is a strongly regular graph: .

• is simply a .

Special cases

• The generalized quadrangles with .

• The Steiner systems with .

Generalisations

A partial linear space is called a semipartial geometry if there are integers such that:

• pairs is incident with .

• 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 .

and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters .

See also

• Strongly regular graph

• Maximal arc

References