Partial geometry
An incidence structure consists of a set {{tmath|1= P }} of points, a set {{tmath|1= L }} of lines, and an incidence relation, or set of flags, ; a point is said to be incident with a line if {{tmath|1= (p,l) \in I }}. It is a (finite) partial geometry if there are integers such that:
- For any pair of distinct points and {{tmath|1= q }}, there is at most one line incident with both of them.
- Each line is incident with points.
- Each point is incident with lines.
- If a point and a line are not incident, there are exactly pairs {{tmath|1= (q,m)\in I}}, such that is incident with and 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 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 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 is simply a {{tmath|1= \mathrm{pg}(t,s,\alpha) }}.
Special cases
- The generalized quadrangles are exactly those partial geometries with {{tmath|1= \alpha=1 }}.
- The Steiner systems are precisely those partial geometries with {{tmath|1= \alpha=s+1 }}.
Generalisations
A partial linear space of order is called a semipartial geometry if there are integers such that:
- If a point and a line are not incident, there are either or exactly pairs {{tmath|1= (q,m)\in I }}, such that is incident with and is incident with {{tmath|1= l}}.
- Every pair of non-collinear points have exactly 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 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
- {{citation |last1=Brouwer |first1=A.E. |last2=van Lint |first2=J.H. |contribution=Strongly regular graphs and partial geometries |editor1-last=Jackson |editor1-first=D.M. |editor2-last=Vanstone |editor2-first=S.A. |title=Enumeration and Design |publisher=Academic Press |place=Toronto |publication-date=1984 |pages=85–122 }}
- {{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 }}
- {{citation |last1=De Clerck |first1=F. |last2=Van Maldeghem |first2=H. |contribution=Some classes of rank 2 geometries |title=Handbook of Incidence Geometry |publisher=North-Holland |place=Amsterdam |publication-date=1995 |pages=433–475 }}
- {{citation |last=Thas |first=J.A. |contribution=Partial Geometries |editor1-last=Colbourn |editor1-first=Charles J. |editor2-last=Dinitz |editor2-first=Jeffrey H. |title=Handbook of Combinatorial Designs |publication-date=2007 |publisher=Chapman & Hall/ CRC |location=Boca Raton |isbn=1-58488-506-8 |edition=2nd |pages=[https://archive.org/details/handbookofcombin0000unse/page/557 557–561] |url=https://archive.org/details/handbookofcombin0000unse/page/557 }}
- {{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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