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Zero-symmetric Graph
In the mathematics, mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique graph automorphism, symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge. The name for this class of graphs was coined by R. M. Foster in a 1966 letter to Harold Scott MacDonald Coxeter, H. S. M. Coxeter. In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.. Examples The smallest zero-symmetric graph is a nonplanar graph with 18 vertices. Its LCF notation is [5,−5]9. Among planar graphs, the truncated cuboctahedral graph, truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric. These examples are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Cuboctahedron
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. Names There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron. Cartesian coordinates The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of: \Bigl(\pm 1, \quad \pm\left(1 + \sqrt 2\right), \quad \pm\left(1 + 2\sqrt 2\right) \Bigr). Area and volume The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge length ''a'' are: :\begin A &= 12\left(2+\sqrt+\sqrt\right) a^2 &&\approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
