The
Pappus graph, a Levi graph with 18 vertices formed from the
Pappus configuration. Vertices labeled with single letters correspond to points in the configuration; vertices labeled with three letters correspond to lines through three points.
The Levi graph of a system of points and lines usually has
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.[1] 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.[4]
Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes in
Euclidean space. For every Levi graph, there is an equivalent
hypergraph, and vice versa.
Examples
Heawood graph and Fano plane Vertex 3 is part of the circular edge (3, 5, 6), the diagonal edge (3, 7, 4), and the side edge (1, 3, 2).
The
Desargues graph is the Levi graph of the
Desargues configuration, 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 is the Levi graph of the
Pappus configuration, 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 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.[5]
References
^
abcGrünbaum, Branko (2006), "Configurations of points and lines", The Coxeter Legacy, Providence, RI: American Mathematical Society, pp. 179–225,
MR2209028. See in particular
p. 181.
^Gropp, Harald (2007), "VI.7 Configurations", in Colbourn, Charles J.; Dinitz, Jeffrey H. (eds.), Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton) (Second ed.), Chapman & Hall/CRC, Boca Raton, Florida, pp. 353–355.