"The" tetrahedral graph is the Platonic graph that is the unique polyhedral graph on four nodes
which is also the complete graph 
and therefore also the wheel
graph 
.
It is implemented in the Wolfram Language
as GraphData["TetrahedralGraph"].
The tetrahedral graph has a single minimal integral embedding, illustrated above (Harborth and Möller 1994), with maximum edge length 4.
The minimal planar integral embedding of the tetrahedral graph, illustrated above, has maximum edge length of 17 (Harborth et al. 1987). The tetrahedral graph is also graceful (Gardner 1983, pp. 158 and 163-164).
The tetrahedral graph has 4 nodes, 6 edges, vertex connectivity 4, edge connectivity 3, graph diameter 1, graph radius 1, and girth 3. It has chromatic polynomial
 |  |  |
(1)
|
 |  |  |
(2)
|
and chromatic number 4. It is planar and cubic symmetric.
The tetrahedral graph is an integral graph with graph spectrum 
. Its automorphism
group has order 
.
The tetrahedral graph is the line graph of the star graph 
,
and the line graph of the tetrahedral graph is the
octahedral graph.
The plots above show the adjacency, incidence, and graph distance matrices for the tetrahedral graph.
The bipartite double graph of the tetrahedral graph is the cubical graph.
The following table summarizes some properties of the tetrahedral graph.
| property | value |
| automorphism group order | 24 |
| characteristic polynomial |  |
| chromatic number | 4 |
| chromatic polynomial |  |
| circulant graph |  |
| claw-free | yes |
| clique number | 4 |
| graph complement name | 4-empty graph |
| determined by spectrum | yes |
| diameter | 1 |
| distance-regular graph | yes |
| dual graph name | tetrahedral graph |
| edge chromatic number | 3 |
| edge connectivity | 3 |
| edge count | 6 |
| Eulerian | no |
| girth | 3 |
| Hamiltonian | yes |
| Hamiltonian cycle count | 6 |
| Hamiltonian path count | 24 |
| integral graph | yes |
| independence number | 1 |
| intersection array |  |
| LCF notation | ![[-2]^4](/images/equations/TetrahedralGraph/Inline16.svg) |
| line graph | yes |
| line graph name | octahedral graph |
| perfect matching graph | no |
| planar | yes |
| polyhedral graph | yes |
| polyhedron embedding names | tetrahedron |
| radius | 1 |
| regular | yes |
| spectrum |  |
| square-free | no |
| strongly regular parameters |  |
| traceable | yes |
| triangle-free | no |
| vertex connectivity | 3 |
| vertex count | 4 |
More generally, a Johnson graph of the from 
(with 
) is known as an 
-tetrahedral graph. For clarity, this work calls such graphs
tetrahedral Johnson graphs.
