An incidence structure C=(P,L,I) consists of a set P of points, a set L of lines, and an incidence relation, or set of flags, I \subseteq P \times L; 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)(s t+\alpha)}{\alpha} and the number of lines by \frac{(t+1)(s t+\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}\Big((s+1)\frac{(s t+\alpha)}{\alpha},s(t+1),s-1+t(\alpha-1),\alpha(t+1)\Big).
  • Partial geometries are dualizable structures: the dual of a \mathrm{pg}(s,t,\alpha) is simply a \mathrm{pg}(t,s,\alpha).

Special cases

  • 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 l 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 l.
  • 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

References