Partial geometry explained

C = (P,L,I) consists a set of points, a set of lines, and an incidence relation, or set of flags, I \subseteq P x L ; a point p is said to be incident with a line l if . 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, 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, such that p is incident with m and q is incident with .

A partial geometry with these parameters is denoted by .

Properties

• The number of points is given by

((s +1 )(s t + \alpha))/(\alpha) and the number of lines by .

• The point graph (also known as the collinearity graph) of a

pg (s,t,\alpha) is a strongly regular graph: .

• Partial geometries are dualizable structures: the dual of a

pg (s,t,\alpha) is simply a .

Special cases

• The generalized quadrangles are exactly those partial geometries

pg (s,t,\alpha) with .

• The Steiner systems

S( 2 , s +1 , ts +1 ) are precisely those partial geometries pg (s,t,\alpha) with .

Generalisations

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, such that p is incident with m and q is incident with .

• Every pair of non-collinear points have exactly

\mu common neighbours.

A semipartial geometry is a partial geometry if and only if .

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

A nice example of such a geometry is obtained by taking the affine points of

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 .

See also

• Strongly regular graph
• Maximal arc