In
combinatorial mathematics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, a Levi graph or incidence graph is a
bipartite graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V a ...
associated with an
incidence structure.
[. See in particula]
p. 181
From a collection of points and lines in an
incidence geometry
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''inciden ...
or a
projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for
Friedrich Wilhelm Levi
Friedrich Wilhelm Daniel Levi (February 6, 1888 – January 1, 1966) was a German mathematician known for his work in abstract algebra, especially torsion-free abelian groups. He also worked in geometry, topology, set theory, and analysis.
Early ...
, who wrote about them in 1942.
The Levi graph of a system of points and lines usually has
girth
Girth may refer to:
;Mathematics
* Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space
* Girth (geometry), the perimeter of a parallel projection of a shape
* Girth ...
at least six: Any 4-
cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure.
Levi graphs of configurations are
biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.
[.]
Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. For every Levi graph, there is an equivalent
hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, an undirected hypergraph H is a pair H = (X,E) w ...
, and vice versa.
Examples
* The
Desargues graph
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high leve ...
is the Levi graph of the
Desargues configuration
In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues.
The Desargues configuration can be constructed in two dimensions ...
, composed of 10 points and 10 lines. There are 3 points on each line, and 3 lines passing through each point. The Desargues graph can also be viewed as the
generalized Petersen graph G(10,3) or the
bipartite Kneser graph with parameters 5,2. It is 3-regular with 20 vertices.
* The
Heawood graph is the Levi graph of the
Fano plane. It is also known as the (3,6)-
cage, and is 3-regular with 14 vertices.
* The
Möbius–Kantor graph is the Levi graph of the
Möbius–Kantor configuration, a system of 8 points and 8 lines that cannot be realized by straight lines in the Euclidean plane. It is 3-regular with 16 vertices.
* The
Pappus graph
In the mathematical field of graph theory, the Pappus graph is a bipartite 3- regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek ...
is the Levi graph of the
Pappus configuration
In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point.
History and construction
This configuration is named after Pappus of A ...
, composed of 9 points and 9 lines. Like the Desargues configuration there are 3 points on each line and 3 lines passing through each point. It is 3-regular with 18 vertices.
* The
Gray graph is the Levi graph of a configuration that can be realized in
as a
grid of 27 points and the 27 orthogonal lines through them.
* The
Tutte eight-cage is the Levi graph of the
Cremona–Richmond configuration. It is also known as the (3,8)-cage, and is 3-regular with 30 vertices.
* The four-dimensional
hypercube graph is the Levi graph of the
Möbius configuration formed by the points and planes of two mutually incident tetrahedra.
* The
Ljubljana graph on 112 vertices is the Levi graph of the Ljubljana configuration.
[.]
References
External links
*
{{Incidence structures
Families of sets
Configurations (geometry)
Geometric graphs