Partial geometry

An incidence structure "S=(P,B,I)" is a (finite) partial geometry if there are integers $s,t,alphageq\; 1$ 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 $alpha$ pairs $(q,M)in\; P\; imes\; B$, such that $p\; I\; M$ , $q\; I\; M$ and $q\; I\; L$.

A partial geometry with these parameters is denoted by $pg(s,t,alpha)$.

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 $pg(s,t,alpha)$ 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 $pg(s,t,alpha)$ is simply a $pg(t,s,alpha)$.

pecial case

* The generalized quadrangles are exactly those partial geometries $pg(s,t,alpha)$ with $alpha=1$.

ee also

* Maximal arc