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 ${\frac {(s+1)(st+\alpha )}{\alpha }}$ and the number of lines by ${\frac {(t+1)(st+\alpha )}{\alpha }}$.
• The point graph (also known as the collinearity graph) of a $\mathrm {pg} (s,t,\alpha )$ is a strongly regular graph: $\mathrm {srg} ((s+1){\frac {(st+\alpha )}{\alpha }},s(t+1),s-1+t(\alpha -1),\alpha (t+1))$.
• Partial geometries are dual structures: the dual of a $\mathrm {pg} (s,t,\alpha )$ 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 $(s,t,\alpha ,\mu )=(q^{2}-1,q^{2}+q,q,q(q+1))$.

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. (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