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 $s, t, α \geq 1$ 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 $α$ 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 ⁠ $p g (s, t, α)$ ⁠.

Properties

The number of points is given by $\frac{(s + 1) (s t + α)}{α}$ and the number of lines by ⁠ $\frac{(t + 1) (s t + α)}{α}$ ⁠.
The point graph (also known as the collinearity graph) of a $p g (s, t, α)$ is a strongly regular graph: ⁠ $s r g ((s + 1) \frac{(s t + α)}{α}, s (t + 1), s - 1 + t (α - 1), α (t + 1))$ ⁠.
Partial geometries are dualizable structures: the dual of a $p g (s, t, α)$ is simply a ⁠ $p g (t, s, α)$ ⁠.

Special cases

The generalized quadrangles are exactly those partial geometries $p g (s, t, α)$ with ⁠ $α = 1$ ⁠.
The Steiner systems $S (2, s + 1, t s + 1)$ are precisely those partial geometries $p g (s, t, α)$ with ⁠ $α = s + 1$ ⁠.

Generalisations

A partial linear space $S = (P, L, I)$ of order $s, t$ is called a semipartial geometry if there are integers $α \geq 1, μ$ such that:

If a point $p$ and a line $l$ are not incident, there are either $0$ or exactly $α$ 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 $μ$ common neighbours.

A semipartial geometry is a partial geometry if and only if ⁠ $μ = α (t + 1)$ ⁠.

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

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

References

Brouwer, A.E.; van Lint, J.H. (1984), "Strongly regular graphs and partial geometries", in Jackson, D.M.; Vanstone, S.A. (eds.), Enumeration and Design, Toronto: Academic Press, pp. 85–122
Bose, R. C. (1963), "Strongly regular graphs, partial geometries and partially balanced designs" (PDF), Pacific J. Math., 13: 389–419, doi:10.2140/pjm.1963.13.389
De Clerck, F.; Van Maldeghem, H. (1995), "Some classes of rank 2 geometries", Handbook of Incidence Geometry, Amsterdam: North-Holland, pp. 433–475
Thas, J.A. (2007), "Partial Geometries", in Colbourn, Charles J.; Dinitz, Jeffrey H. (eds.), Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, pp. 557–561, ISBN 1-58488-506-8
Debroey, I.; Thas, J. A. (1978), "On semipartial geometries", Journal of Combinatorial Theory, Series A, 25: 242–250, doi:10.1016/0097-3165(78)90016-x

Properties

Special cases

Generalisations

See also

References