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# Great Rhombicuboctahedral Graph

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

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

It has graph spectrum

where , , and are roots of and , , and are roots of .

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.

Archimedean Graph, Great Rhombicuboctahedron, Zero-Symmetric Graph

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## 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