Partial geometry

An incidence structure $C=(P,L,I)$ consists of a set of points, a set of lines, and an incidence relation, or set of flags, $I \subseteq P \times L$; a point $p$ is said to be ''incident'' with a line $l$ if . It is a ( finite) partial geometry if there are integers $s,t,\alpha\geq 1$ such that:
* For any pair of distinct points $p$ and , there is at most one line incident with both of them.
* Each line is incident with $s+1$ points.
* Each point is incident with $t+1$ lines.
* If a point $p$ and a line $l$ are not incident, there are exactly $\alpha$ pairs , such that $p$ is incident with $m$ and $q$ is incident with .

A partial geometry with these parameters is denoted by .

Properties

* The number of points is given by $\frac$ and the number of lines by .
* The point graph (also known as the collinearity graph) of a $\mathrm(s,t,\alpha)$ is a strongly regular graph: .
* Partial geometries are dualizable structures: the dual of a $\mathrm(s,t,\alpha)$ is simply a .

Special cases

* The generalized quadrangles are exactly those partial geometries $\mathrm(s,t,\alpha)$ with .
* The Steiner systems $S(2, s+1, ts+1)$ are precisely those partial geometries $\mathrm(s,t,\alpha)$ with .

Generalisations

A partial linear space $S=(P,L,I)$ of order $s, t$ is called a semipartial geometry if there are integers $\alpha\geq 1, \mu$ such that:
* If a point $p$ and a line $l$ are not incident, there are either $0$ or exactly $\alpha$ pairs , such that $p$ is incident with $m$ and $q$ is incident with .
* Every pair of non-collinear points have exactly $\mu$ 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 $\mathrm(3, q^2)$ 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

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{{DEFAULTSORT:Partial Geometry
Incidence geometry