Partial geometry
Encyclopedia
An incidence structure
consists of points 
, lines 
, and flags 
where a point 
is said to be incident with a line 
if 
. It is a (finite) partial geometry if there are integer
s
such that:
A partial geometry with these parameters is denoted by
.
Incidence structure
In mathematics, an incidence structure is a tripleC=.\,where P is a set of "points", L is a set of "lines" and I \subseteq P \times L is the incidence relation. The elements of I are called flags. If \in I,...
consists of points , lines , and flags where a point is said to be incident with a line if . It is a (finite) partial geometry if there are integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s
such that:- For any pair of distinct points
and 
, 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 
, such that 
is incident with 
and 
is incident with 
.
A partial geometry with these parameters is denoted by
.Properties
- The number of points is given by
and the number of lines by 
. - The point graph of a
is a strongly regular graphStrongly regular graphIn graph theory, a discipline within mathematics, a strongly regular graph is defined as follows. Let G = be a regular graph with v vertices and degree k...
:
. - Partial geometries are dual structures : the dual of a
is simply a 
.
Special case
- The generalized quadrangleGeneralized quadrangleA generalized quadrangle is an incidence structure. A generalized quadrangle is by definition a polar space of rank two. They are the generalized n-gons with n=4...
s are exactly those partial geometries
with 
.