Partial geometry

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An incidence structure S=(P,B,I) is a (finite) partial geometry if there are integers $s,t,\alpha\geq 1$ such that:

For each two different points p and q, 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 $α$ pairs $(q,M)\in P\times B$ , such that $p I M$ , $q I M$ and $q I L$ .

A partial geometry with these parameters is denoted by $p g (s, t,α)$ .

[edit] Properties

The number of points is given by $\frac{(s+1)(s t+\alpha)}{\alpha}$ and the number of lines by $\frac{(t+1)(s t+\alpha)}{\alpha}$ .
The point graph of a $p g (s, t,α)$ is a strongly regular graph : $srg((s+1)\frac{(s t+\alpha)}{\alpha},s(t+1),s-1+t(\alpha-1),\alpha(t+1))$ .
Partial geometries are dual structures : the dual of a $p g (s, t,α)$ is simply a $p g (t, s,α)$ .

[edit] Special case

The generalized quadrangles are exactly those partial geometries $p g (s, t,α)$ with $α = 1$ .

[edit] See also

Maximal arc

Categories: Incidence geometry

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