Partial geometry

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

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

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

Properties

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

Special case

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

Generalisations

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

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

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

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

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

See also

References



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