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Great Rhombicuboctahedral Graph


GreatRhombicuboctahedralGraph

The great rhombicuboctahedral graph is the cubic Archimedean graph on 48 nodes and 72 edges that is the skeleton of the great rhombicuboctahedron as well as the great truncated cuboctahedron and quasirhombicuboctahedron uniform polyhedra.

It is implemented in the Wolfram Language as GraphData["GreatRhombicuboctahedralGraph"].

It has chromatic number 2, vertex connectivity 3, edge connectivity 3, graph diameter 9, graph radius 9, and girth 4. It is cubic, planar, and Hamiltonian. It is also zero-symmetric

GreatRhombicuboctahedralGraphLCF

It is Hamiltonian with 2684 Hamiltonian cycles. It has 37 distinct LCF notations, one of order 4 ([-11,11,-3,7,5,-9,-11,11,9,-5,-7,3]^4), one of order 3 ([-15, 15, -3, -7, -9, 9, 7, -13, -15, 15, 13, 9, 7, -7, -9, 3]^3), eight of order 2, and 27 of order 1. The first of these are illustrated above.

It has graph spectrum

 (-3)^1(-1-sqrt(3))^3(lambda,mu,nu)^3(-2)^2(-1)^4(1-sqrt(3))^3×0^4(s,t,u)^3(-1+sqrt(3))^31^42^23^1,

where lambda, mu, and nu are roots of x^3+x^2-4x-2 and s, t, and u are roots of x^3+x^2-4x-2.

It is the Cayley graph of the permutations {{1, 2, 3, 4, 5, 7, 6}, {1, 2, 3, 4, 6, 5, 7}, {1, 3, 2, 5, 4, 7, 6}}.


See also

Archimedean Graph, Great Rhombicuboctahedron, Zero-Symmetric Graph

Explore with Wolfram|Alpha

References

Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 268, 1998.

Cite this as:

Weisstein, Eric W. "Great Rhombicuboctahedral Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreatRhombicuboctahedralGraph.html

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