The rhombic triacontahedral graph is an Archimedean dual graph which is the skeleton of the rhombic triacontahedron, great rhombic triacontahedron, and small triambic icosahedron.
The rhombic triacontahedral graph is also its own distance-5 graph distance graph.
It is implemented in the Wolfram Language as GraphData["RhombicTriacontahedralGraph"].
The plots above show the adjacency matrix, incidence matrix, and graph distance matrix for the rhombic triacontahedral graph.
The following table summarizes some properties of the graph.
| property | value |
| automorphism group order | 120 |
| characteristic polynomial |  |
| chromatic number | 2 |
| chromatic polynomial | ? |
| claw-free | no |
| clique number | 2 |
| determined by spectrum | ? |
| diameter | 6 |
| distance-regular graph | no |
| dual graph name | icosidodecahedral graph |
| edge chromatic number | 5 |
| edge connectivity | 3 |
| edge count | 60 |
| Eulerian | no |
| girth | 4 |
| Hamiltonian | no |
| Hamiltonian cycle count | 0 |
| Hamiltonian path count | 0 |
| integral graph | no |
| independence number | 20 |
| line graph | ? |
| perfect matching graph | no |
| planar | yes |
| polyhedral graph | yes |
| polyhedron embedding names | rhombic triacontahedron |
| radius | 6 |
| regular | no |
| square-free | no |
| traceable | no |
| triangle-free | yes |
| vertex connectivity | 3 |
| vertex count | 32 |
