Levi graph

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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.

In combinatorics a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. 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. Any Levi graph has girth at least six: there can be no 4-cycles, because that 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. For every Levi graph, there is an equivalent hypergraph, and vice versa.

[edit] Examples

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.