Partial geometry

An incidence structure $C=(P,L,I)$ consists of a set Template:Tmath of points, a set Template:Tmath 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 Template:Tmath. It is a (finite) partial geometry if there are integers $s,t,\alpha \ge 1$ such that:

• For any pair of distinct points $p$ and Template:Tmath, 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 Template:Tmath, such that $p$ is incident with $m$ and $q$ is incident with Template:Tmath.

A partial geometry with these parameters is denoted by Template:Tmath.

Properties

• The number of points is given by $\frac{(s+1)(st+\alpha )}{\alpha }$ and the number of lines by Template:Tmath.
• The point graph (also known as the collinearity graph) of a $\mathrm{p}\mathrm{g}(s,t,\alpha )$ is a strongly regular graph: Template:Tmath.
• Partial geometries are dualizable structures: the dual of a $\mathrm{p}\mathrm{g}(s,t,\alpha )$ is simply a Template:Tmath.

Special cases

• The generalized quadrangles are exactly those partial geometries $\mathrm{p}\mathrm{g}(s,t,\alpha )$ with Template:Tmath.
• The Steiner systems $S(2,s+1,ts+1)$ are precisely those partial geometries $\mathrm{p}\mathrm{g}(s,t,\alpha )$ with Template:Tmath.

Generalisations

A partial linear space $S=(P,L,I)$ of order $s,t$ is called a semipartial geometry if there are integers $\alpha \ge 1,\mu$ such that:

• If a point $p$ and a line $l$ are not incident, there are either $0$ or exactly $\alpha$ pairs Template:Tmath, such that $p$ is incident with $m$ and $q$ is incident with Template:Tmath.
• Every pair of non-collinear points have exactly $\mu$ common neighbours.

A semipartial geometry is a partial geometry if and only if Template:Tmath.

It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters Template:Tmath.

A nice example of such a geometry is obtained by taking the affine points of $\mathrm{P}\mathrm{G}(3,q^2)$ and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters Template:Tmath.

See also

• Strongly regular graph
• Maximal arc

References

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