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 $s,t,\alpha \ge 1$ 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 $\mathrm{p}\mathrm{g}(s,t,\alpha )$.

Properties

• 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 $\mathrm{p}\mathrm{g}(s,t,\alpha )$ is a strongly regular graph: $\mathrm{s}\mathrm{r}\mathrm{g}((s+1)\frac{(st+\alpha )}{\alpha },s(t+1),s-1+t(\alpha -1),\alpha (t+1))$.
• Partial geometries are dual structures: the dual of a $\mathrm{p}\mathrm{g}(s,t,\alpha )$ is simply a $\mathrm{p}\mathrm{g}(t,s,\alpha )$.

Special case

• The generalized quadrangles are exactly those partial geometries $\mathrm{p}\mathrm{g}(s,t,\alpha )$ with $\alpha =1$.
• The Steiner systems $S(2,s+1,ts+1)$ are precisely those partial geometries $\mathrm{p}\mathrm{g}(s,t,\alpha )$ 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 $\alpha \ge 1,\mu$ 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 $\mathrm{P}\mathrm{G}(3,q^2)$ and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters $(s,t,\alpha ,\mu )=(q^2-1,q^2+q,q,q(q+1))$.

See also

• Strongly regular graph
• Maximal arc

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 
• Bose, R. C. (1963), "Strongly regular graphs, partial geometries and partially balanced designs", Pacific J. Math. 13: 389–419, doi:10.2140/pjm.1963.13.389, https://projecteuclid.org/euclid.pjm/1103035734 
• 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, https://archive.org/details/handbookofcombin0000unse/page/557 
• 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