It can be defined as the generalized Petersen graph G(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it (an octagram).

For p = 4 there is no solution in the Euclidean plane, but Kantor (1882) found pairs of polygons of this type, for a generalization of the problem in which the points and edges belong to the complex projective plane.

That is, in Kantor's solution, the coordinates of the polygon vertices are complex numbers.

Kantor's solution for p = 4, a pair of mutually-inscribed quadrilaterals in the complex projective plane, is called the Möbius–Kantor configuration.

[1] Since the hypercube is a unit distance graph, the Möbius–Kantor graph can also be drawn in the plane with all edges unit length, although such a drawing will necessarily have some pairs of crossing edges.

Each of these instances is in fact an eigenvector of the Hoffman-Singleton graph, with associated eigenvalue -3.

[2] Additionally, it provides an example of a graph all of whose subgraphs' crossing numbers differ from it by two or more.

[3] However, it is a toroidal graph: it has an embedding in the torus in which all faces are hexagons.

There is an even more symmetric embedding of Möbius–Kantor graph in the double torus which is a regular map, with six octagonal faces, in which all 96 symmetries of the graph can be realized as symmetries of the embedding[5] Its 96-element symmetry group has a Cayley graph that can itself be embedded on the double torus, and was shown by Tucker (1984) to be the unique group with genus two.

This means it can be obtained from the regular map as a skeleton of the dual of its barycentric subdivision.

A sculpture by DeWitt Godfrey and Duane Martinez showing the double torus embedding of the symmetries of the Möbius–Kantor graph was unveiled at the Technical Museum of Slovenia as part of the 6th Slovenian International Conference on Graph Theory in 2007.

[4] Lijnen & Ceulemans (2004), motivated by an investigation of potential chemical structures of carbon compounds, studied the family of all embeddings of the Möbius–Kantor graph onto 2-manifolds; they showed that there are 759 inequivalent embeddings.

[4] It acts transitively on the vertices, on the edges and on the arcs of the graph.

The generalized Petersen graph G(n,k) is vertex-transitive if and only if n = 10 and k =2 or if k2 ≡ ±1 (mod n) and is edge-transitive only in the following seven cases: (n,k) = (4,1), (5,2), (8,3), (10,2), (10,3), (12,5), or (24,5).