Levi Graph - Examples

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.

• 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 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 Q₄ 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.