The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, illustrated above in four embeddings. The left embedding shows a stereographic projection of the dodecahedron, the second an orthographic projection, the third is from Read and Wilson (1998, p. 162), and the fourth is derived from LCF notation.
The dodecahedral graph is the skeleton of the great stellated dodecahedron as well as the dodecahedron.
It is the cubic symmetric denoted 
and is isomorphic to the generalized
Petersen graph 
.
It can be described in LCF notation as [10, 7, 4,
,
,
10, 
,
7, 
,
![4]^2](/images/equations/DodecahedralGraph/Inline7.svg)
.
The dodecahedral graph is implemented in the Wolfram Language as GraphData["DodecahedralGraph"].
It is distance-regular with intersection array 
and is also distance-transitive.
It is also a unit-distance graph (Gerbracht 2008), as shown above in a unit-distance embedding.
Finding a Hamiltonian cycle on this graph is known as the icosian game. The dodecahedral graph
is not Hamilton-connected and is the
only known example of a vertex-transitive Hamiltonian graph (other than cycle
graphs 
)
that is not H-*-connected (Stan Wagon,
pers. comm., May 20, 2013).
The dodecahedral graph has 20 nodes, 30 edges, vertex connectivity 3, edge connectivity 3, graph diameter 5, graph
radius 5, and girth 5. Its has chromatic
number 3. Its graph spectrum is 
(Buekenhout and
Parker 1998; Cvetkovic et al. 1998, p. 308). Its automorphism
group is of order 
(Buekenhout and Parker 1998).
The minimal planar integral embedding of the dodecahedral graph has maximum edge length of 2 (Harborth et al. 1987). It
is also graceful (Gardner 1983, pp. 158 and
163-164; Gallian 2018, p. 35) with 
fundamentally different labelings, giving a total
number of 
graceful labelings (B. Dobbelaere, pers. comm., Oct. 22, 2020), a number
independently (and nearly simultaneously!) determined by T. Rokicki on Oct. 6,
2020 (D. Knuth, pers. comm., Jul. 6, 2023).
The dodecahedral graph can be constructed as the graph expansion of 
with steps 1 and 2, where 
is a path graph (Biggs 1993,
p. 119).
The skeleton of the great stellated dodecahedron is isomorphic to the dodecahedral graph.
The line graph of the dodecahedral graph is the icosidodecahedral graph. The graph square of the dodecahedral graph is the crossed dodecahedral graph.
The dodecahedral graph has chromatic polynomial
 |
The plots above show the adjacency, incidence, and graph distance matrices for the dodecahedral graph.
The bipartite double graph of the dodecahedral graph is the cubic symmetric graph 
.
The following table summarizes properties of the dodecahedral graph.
| property | value |
| automorphism group order | 120 |
| characteristic polynomial |  |
| chromatic number | 3 |
| chromatic polynomial |  |
| claw-free | no |
| clique number | 2 |
| determined by spectrum | yes |
| diameter | 5 |
| distance-regular graph | yes |
| dual graph name | icosahedral graph |
| edge chromatic number | 3 |
| edge connectivity | 3 |
| edge count | 30 |
| Eulerian | no |
| generalized Petersen indices |  |
| girth | 5 |
| Hamiltonian | yes |
| Hamiltonian cycle count | 60 |
| Hamiltonian path count | ? |
| integral graph | no |
| independence number | 8 |
| LCF notation | ![[-10,-4,7,-7,4,-10,7,4,-4,-7]^2](/images/equations/DodecahedralGraph/Inline20.svg) |
| line graph | ? |
| line graph name | icosidodecahedral graph |
| perfect matching graph | no |
| planar | yes |
| polyhedral graph | yes |
| polyhedron embedding names | dodecahedron, great stellated dodecahedron |
| radius | 5 |
| regular | yes |
| spectrum |  |
| square-free | yes |
| traceable | yes |
| triangle-free | yes |
| vertex connectivity | 3 |
| vertex count | 20 |
| weakly regular parameters |  |
."
Disc. Math. 186, 69-94, 1998.Chartrand, G.