In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3-vertex-connected, vertex-transitive, and planar graphs), and also Hamiltonian graphs.
Examples
The first graph in the sequence, the octahedral graph, has 6 vertices and 12 edges. Later graphs in the sequence may be named after the type of antiprism they correspond to:
- Octahedral graph – 6 vertices, 12 edges
- square antiprismatic graph – 8 vertices, 16 edges
- Pentagonal antiprismatic graph – 10 vertices, 20 edges
- Hexagonal antiprismatic graph – 12 vertices, 24 edges
- Heptagonal antiprismatic graph – 14 vertices, 28 edges
- Octagonal antiprismatic graph – 16 vertices, 32 edges
- ...
 3 |
 4 |
 5 |
 6 |
 7 |
 8 |
Although geometrically the star polygons also form the faces of a different sequence of (self-intersecting) antiprisms, the star antiprisms, they do not form a different sequence of graphs.
Related graphs
An antiprism graph is a special case of a circulant graph, Ci2n(2,1).
Other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs.
References
- Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 270.