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 $\alpha=s+1$.

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 $PG(3, q^2)$ 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

• 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
• 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
• Debroey, I.; Thas, J. A. (1978), "On semipartial geometries", Journal of Combinatorial Theory Ser. A 25: 242–250