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 $\frac{(s+1)(s t+\alpha)}{\alpha}$ and the number of lines by $\frac{(t+1)(s t+\alpha)}{\alpha}$ .
The point graph of a $pg(s,t,\alpha)$ is a strongly regular graph : $srg((s+1)\frac{(s t+\alpha)}{\alpha},s(t+1),s-1+t(\alpha-1),\alpha(t+1))$ .
Partial geometries are dual structures : the dual of a $pg(s,t,\alpha)$ 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 are precisely those partial geometries $pg(s,t,\alpha)$ with $\alpha=s+1$ .

References

Brouwer, A.E.; van Lint, J.H. (1984), "Strongly regular graphs and partial geometries", in Jackson, D.M.; Vanstone, S.A., Enumeration and Design, Toronto: Academic Press, pp. 85–122
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., Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, pp. 557–561, ISBN 1-58488-506-8

Partial geometry

Properties

Special case

See also

References