Partial geometry

An incidence structure ${\displaystyle C=(P,L,I)}$ consists of a set ⁠${\displaystyle P}$⁠ of points, a set ⁠${\displaystyle L}$⁠ of lines, and an incidence relation, or set of flags, ${\displaystyle I\subseteq P\times L}$; a point ${\displaystyle p}$ is said to be incident with a line ${\displaystyle l}$ if ⁠${\displaystyle (p,l)\in I}$⁠. It is a (finite) partial geometry if there are integers ${\displaystyle s,t,\alpha \geq 1}$ such that:

• For any pair of distinct points ${\displaystyle p}$ and ⁠${\displaystyle q}$⁠, there is at most one line incident with both of them.
• Each line is incident with ${\displaystyle s+1}$ points.
• Each point is incident with ${\displaystyle t+1}$ lines.
• If a point ${\displaystyle p}$ and a line ${\displaystyle l}$ are not incident, there are exactly ${\displaystyle \alpha }$ pairs ⁠${\displaystyle (q,m)\in I}$⁠, such that ${\displaystyle p}$ is incident with ${\displaystyle m}$ and ${\displaystyle q}$ is incident with ⁠${\displaystyle l}$⁠.

A partial geometry with these parameters is denoted by ⁠${\displaystyle \mathrm {pg} (s,t,\alpha )}$⁠.

Properties

• The number of points is given by ${\displaystyle {\frac {(s+1)(st+\alpha )}{\alpha }}}$ and the number of lines by ⁠${\displaystyle {\frac {(t+1)(st+\alpha )}{\alpha }}}$⁠.
• The point graph (also known as the collinearity graph) of a ${\displaystyle \mathrm {pg} (s,t,\alpha )}$ is a strongly regular graph: ⁠${\displaystyle \mathrm {srg} {\Big (}(s+1){\frac {(st+\alpha )}{\alpha }},s(t+1),s-1+t(\alpha -1),\alpha (t+1){\Big )}}$⁠.
• Partial geometries are dualizable structures: the dual of a ${\displaystyle \mathrm {pg} (s,t,\alpha )}$ is simply a ⁠${\displaystyle \mathrm {pg} (t,s,\alpha )}$⁠.

Special cases

• The generalized quadrangles are exactly those partial geometries ${\displaystyle \mathrm {pg} (s,t,\alpha )}$ with ⁠${\displaystyle \alpha =1}$⁠.
• The Steiner systems ${\displaystyle S(2,s+1,ts+1)}$ are precisely those partial geometries ${\displaystyle \mathrm {pg} (s,t,\alpha )}$ with ⁠${\displaystyle \alpha =s+1}$⁠.

Generalisations

A partial linear space ${\displaystyle S=(P,L,I)}$ of order ${\displaystyle s,t}$ is called a semipartial geometry if there are integers ${\displaystyle \alpha \geq 1,\mu }$ such that:

• If a point ${\displaystyle p}$ and a line ${\displaystyle l}$ are not incident, there are either ${\displaystyle 0}$ or exactly ${\displaystyle \alpha }$ pairs ⁠${\displaystyle (q,m)\in I}$⁠, such that ${\displaystyle p}$ is incident with ${\displaystyle m}$ and ${\displaystyle q}$ is incident with ⁠${\displaystyle l}$⁠.
• Every pair of non-collinear points have exactly ${\displaystyle \mu }$ common neighbours.

A semipartial geometry is a partial geometry if and only if ⁠${\displaystyle \mu =\alpha (t+1)}$⁠.

It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters ⁠${\displaystyle (1+s(t+1)+s(t+1)t(s-\alpha +1)/\mu ,s(t+1),s-1+t(\alpha -1),\mu )}$⁠.

A nice example of such a geometry is obtained by taking the affine points of ${\displaystyle \mathrm {PG} (3,q^{2})}$ and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters ⁠${\displaystyle (s,t,\alpha ,\mu )=(q^{2}-1,q^{2}+q,q,q(q+1))}$⁠.

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