Partial geometry

An incidence structure $C=(P,L,I)$ consists of points $P$, lines $L$, and flags $I\subseteq P\times L$ where a point $p$ is said to be incident with a line $l$ if $(p,l)\in I$. 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 $q$, 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 $(q,m)\in I$, such that $p$ is incident with $m$ and $q$ is incident with $l$.

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

Properties

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

Special case

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

See also

• Maximal arc

References

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