Partial geometry
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 (also known as the collinearity 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
 Strongly regular graph
 Maximal arc

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Incidence geometry