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 $\mathrm{pg}(s,t,\alpha)$.

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

Special case

 * 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 $$\ell$$ 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 $$\mathrm{PG}(3, q^2)$$ and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters $(s, t, \alpha, \mu) = (q^2 - 1, q^2 + q, q, q(q + 1))$.