Schläfli graph

That is, two vertices are adjacent in the Schläfli graph if and only if the corresponding pair of lines are skew.

It plays an important role in the structure theory for claw-free graphs by Chudnovsky & Seymour (2005).

Correspondingly, in the Schläfli graph, each edge uv belongs uniquely to a subgraph in the form of a Cartesian product of complete graphs K6

The Schläfli graph has a total of 36 subgraphs of this form, one of which consists of the zero-one vectors in the eight-dimensional representation described above.

The proof relies on the classification of finite simple groups.

The Schläfli graph is seen as a 1-skeleton of the 2 21 polytope . This symmetric projection contains 2 rings of 12 vertices, and 3 vertices coinciding at the center.