An incidence structure C = ( P , L , I ) consists of points P , lines L , and flags I ⊆ P × L where a point p is said to be incident with a line l if ( p , l ) ∈ I . It is a (finite) partial geometry if there are integers s , t , α ≥ 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 α pairs ( q , m ) ∈ I , such that p is incident with m and q is incident with l .A partial geometry with these parameters is denoted by p g ( s , t , α ) .
The number of points is given by ( s + 1 ) ( s t + α ) α and the number of lines by ( t + 1 ) ( s t + α ) α .The point graph of a p g ( s , t , α ) is a strongly regular graph : s r g ( ( s + 1 ) ( s t + α ) α , s ( t + 1 ) , s − 1 + t ( α − 1 ) , α ( t + 1 ) ) .Partial geometries are dual structures : the dual of a p g ( s , t , α ) is simply a p g ( t , s , α ) .The generalized quadrangles are exactly those partial geometries p g ( s , t , α ) with α = 1 .The Steiner systems are precisely those partial geometries p g ( s , t , α ) with α = s + 1 .A partial linear space S = ( P , L , I ) of order s , t is called a semipartial geometry if there are integers α ≥ 1 , μ such that:
If a point p and a line ℓ are not incident, there are either 0 or exactly α pairs ( q , m ) ∈ I , such that p is incident with m and q is incident with ℓ .Every pair of non-collinear points have exactly μ common neighbours.A semipartial geometry is a partial geometry if and only if μ = α ( 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 − α + 1 ) / μ , s ( t + 1 ) , s − 1 + t ( α − 1 ) , μ ) .
A nice example of such a geometry is obtained by taking the affine points of P G ( 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 , α , μ ) = ( q 2 − 1 , q 2 + q , q , q ( q + 1 ) ) .
