Partial geometry

An incidence structure C=(P,L,I) consists of a set {{tmath|1= P }} of points, a set {{tmath|1= 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 {{tmath|1= (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 {{tmath|1= 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 {{tmath|1= (q,m)\in I}}, such that p is incident with m and q is incident with {{tmath|1= l }}.

A partial geometry with these parameters is denoted by {{tmath|1= \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 {{tmath|1= \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: {{tmath|1= \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 {{tmath|1= \mathrm{pg}(t,s,\alpha) }}.

Special cases

  • The generalized quadrangles are exactly those partial geometries \mathrm{pg}(s,t,\alpha) with {{tmath|1= \alpha=1 }}.
  • The Steiner systems S(2, s+1, ts+1) are precisely those partial geometries \mathrm{pg}(s,t,\alpha) with {{tmath|1= \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 {{tmath|1= (q,m)\in I }}, such that p is incident with m and q is incident with {{tmath|1= l}}.
  • Every pair of non-collinear points have exactly \mu common neighbours.

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

It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters

{{tmath|1= (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 {{tmath|1= (s, t, \alpha, \mu) = (q^2 - 1, q^2 + q, q, q(q + 1)) }}.

See also

References

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  • {{citation |last = Bose |first = R. C. |title = Strongly regular graphs, partial geometries and partially balanced designs |journal = Pacific J. Math. |volume = 13 |date = 1963 |url = https://repository.lib.ncsu.edu/bitstream/1840.4/2478/1/ISMS_1963_358.pdf |pages = 389–419 |doi=10.2140/pjm.1963.13.389 |doi-access = free }}
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  • {{citation |last1=Debroey |first1=I. |last2=Thas |first2=J. A. |title=On semipartial geometries |journal=Journal of Combinatorial Theory, Series A |publication-date=1978 |volume=25 |pages=242–250 |doi=10.1016/0097-3165(78)90016-x |doi-access=free }}

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Category:Incidence geometry