Bidiakis cube
{{Short description|3-regular graph with 12 vertices and 18 edges}}
{{infobox graph
| name = bidiakis cube
| image = 220px
| image_caption = The bidiakis cube
| namesake =
| vertices = 12
| edges = 18
| automorphisms = 8 (D4)
| girth = 4
| diameter = 3
| radius = 3
| chromatic_number = 3
| chromatic_index = 3
| properties = Cubic
Hamiltonian
Triangle-free
Polyhedral
Planar
}}
In the mathematical field of graph theory, the bidiakis cube is a 3-regular graph with 12 vertices and 18 edges.{{MathWorld|urlname=BidiakisCube|title=Bidiakis cube}}
Construction
The bidiakis cube is a cubic Hamiltonian graph and can be defined by the LCF notation [−6,4,−4]4.
The bidiakis cube can also be constructed from a cube by adding edges across the top and bottom faces which connect the centres of opposite sides of the faces. The two additional edges need to be perpendicular to each other. With this construction, the bidiakis cube is a polyhedral graph, and can be realized as a convex polyhedron. Therefore, by Steinitz's theorem, it is a 3-vertex-connected simple planar graph.Branko Grünbaum, Convex Polytopes, 2nd edition, prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler, 2003, {{ISBN|0-387-40409-0}}, {{ISBN|978-0-387-40409-7}}, 466pp.
Algebraic properties
The bidiakis cube is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 8, the group of symmetries of a square, including both rotations and reflections.
The characteristic polynomial of the bidiakis cube is .
Gallery
Image:Bidiakis cube 3COL.svg|The chromatic number of the bidiakis cube is 3.
Image:Bidiakis cube 3color edge.svg|The chromatic index of the bidiakis cube is 3.
Image:Bidiakis cube planar.svg|The bidiakis cube is a planar graph.
Image:Bidiakis cube.svg|The bidiakis cube constructed from a cube.