The 1 -skeleton of a polytope can be thought of as a graph with its vertex set being the vertices of the polytope and its edge set being the edges of the polytope. Thus two vertices are adjacent iff there is an edge in the polytope which is incident on both of them. This graph may be referenced by a number of names. Since the 1 -skeleton is the most commonly used k -skeleton it may be referred to as "the skeleton" of a polytope without a qualifying numeral. It may also be called simply the graph of a polytope or the vertex adjacency graph of a polytope.
Properties[edit | edit source]
Polytopes of different ranks can have the same 1-skeleton. The complete graph is the 1-skeleton of both a 4-polytope and 5-polytope.
References[edit | edit source]
Bibligraphy[edit | edit source]
- Kalai, Gil (10 August 2017). "Polytope Skeletons and Paths" (PDF).
- Steinitz, Ernst (1922). "Polyeder und Raumeinteilungen" [Polyhedra and Divisions of Space]. Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries) [Encyclopedia of the Mathematical Sciences, Volume 3 (Geometry)] (in German).