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 $\frac$ and the number of lines by $\frac$.
* The point graph (also known as the collinearity graph) of a $pg(s,t,\alpha)$ is a strongly regular graph: $srg((s+1)\frac,s(t+1),s-1+t(\alpha-1),\alpha(t+1))$.
* 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 $\alpha=1$.
* The Steiner systems $S(2, s+1, ts+1)$ are precisely those partial geometries $pg(s,t,\alpha)$ 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 $\ell$ 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 $\ell$.
* 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 $PG(3, q^2)$ and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters $(s, t, \alpha, \mu) = (q^2 - 1, q^2 + q, q, q(q + 1))$.

See also

* Strongly regular graph
* Maximal arc

References

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{{DEFAULTSORT:Partial Geometry
Incidence geometry