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

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

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Partial geometry

Properties

Special case

Generalisations

See also

References