The great rhombicuboctahedral graph is the cubic Archimedean graph on 48 vertices 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 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](/images/equations/GreatRhombicuboctahedralGraph/Inline1.svg)
), one of order 3 (![[-15,15,-3,-7,-9,9,7,-13,-15,15,13,9,7,-7,-9,3]^3](/images/equations/GreatRhombicuboctahedralGraph/Inline2.svg)
),
eight of order 2, and 27 of order 1. The first of these is 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
.
