Partial geometry

An incidence structure $C=(P,L,I)$ consists of a set ⁠ $P$ ⁠ of points, a set ⁠ $L$ ⁠ 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 ⁠ $(p,l)\in I$ ⁠. 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 ⁠ $\mathrm {pg} (s,t,\alpha )$ ⁠.

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 (also known as the collinearity graph) of a $\mathrm {pg} (s,t,\alpha )$ is a strongly regular graph: ⁠ $\mathrm {srg} {\Big (}(s+1){\frac {(st+\alpha )}{\alpha }},s(t+1),s-1+t(\alpha -1),\alpha (t+1){\Big )}$ ⁠.
Partial geometries are dualizable structures: the dual of a $\mathrm {pg} (s,t,\alpha )$ is simply a ⁠ $\mathrm {pg} (t,s,\alpha )$ ⁠.

Special cases

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

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 ⁠ $(q,m)\in I$ ⁠, such that $p$ is incident with $m$ and $q$ is incident with ⁠ $l$ ⁠.
Every pair of non-collinear points have exactly $\mu$ common neighbours.

A semipartial geometry is a partial geometry if and only if ⁠ $\mu =\alpha (t+1)$ ⁠.

It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters ⁠ $(1+s(t+1)+s(t+1)t(s-\alpha +1)/\mu ,s(t+1),s-1+t(\alpha -1),\mu )$ ⁠.

A nice example of such a geometry is obtained by taking the affine points of $\mathrm {PG} (3,q^{2})$ and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters ⁠ $(s,t,\alpha ,\mu )=(q^{2}-1,q^{2}+q,q,q(q+1))$ ⁠.

Special cases

Generalisations

See also

References

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