Levi graph

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Levi graph
Levi graph of Pappus Configuration.png
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.
Girth ≥ 6

In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.[1][2] From a collection of points and lines in an incidence geometry 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 F. W. Levi, who wrote about them in 1942.[1][3]

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[edit]

  • 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 R3 as a 3×3×3 grid of 27 points and the 27 orthogonal lines through them.
  • The Ljubljana graph on 112 vertices is the Levi graph of the Ljubljana configuration.[5]

References[edit]

  1. ^ a b c Grünbaum, Branko (2006), "Configurations of points and lines", The Coxeter Legacy, Providence, RI: American Mathematical Society, pp. 179–225, MR 2209028 . See in particular p. 181.
  2. ^ Polster, Burkard (1998), A Geometrical Picture Book, Universitext, New York: Springer-Verlag, p. 5, doi:10.1007/978-1-4419-8526-2, ISBN 0-387-98437-2, MR 1640615 .
  3. ^ Levi, F. W. (1942), Finite Geometrical Systems, Calcutta: University of Calcutta, MR 0006834 .
  4. ^ Gropp, Harald (2007), "VI.7 Configurations", in Colbourn, Charles J.; Dinitz, Jeffrey H., Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton) (Second ed.), Chapman & Hall/CRC, Boca Raton, FL, pp. 353–355 .
  5. ^ Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; Potočnik, P. (2002), The Ljubljana Graph, IMFM Preprint 40-845, University of Ljubljana Department of Mathematics .

External links[edit]