The icosahedral graph is the Platonic graph whose nodes have the connectivity of the regular icosahedron, as well as the great dodecahedron, great icosahedron, Jessen's orthogonal icosahedron, and small stellated dodecahedron. The icosahedral graph has 12 vertices and 30 edges and is illustrated above in a number of drawings.
Since the icosahedral graph is regular and Hamiltonian, it has a generalized LCF notation. In fact, there
are two distinct generalized LCF notations of order
6--![[(-4,-3,4),(-2,2,3)]^6](/images/equations/IcosahedralGraph/Inline1.svg)
and ![[(-4,3,4),(-3,-2,2)]^6](/images/equations/IcosahedralGraph/Inline2.svg)
--8
of order 2, and 17 of order 1, illustrated above.
It is implemented in the Wolfram Language as GraphData["IcosahedralGraph"].
It is a distance-regular graph with intersection array 
, and therefore also a Taylor
graph. It is also distance-transitive.
It is its own distance-2 graph distance graph.
The icosahedral graph is graceful (Gardner 1983, pp. 158 and 163-164; Gallian 2025; Knuth 2025), as shown by the labeling above which gives absolute values of differences of adjacent labeled vertices consisting of precisely the numbers 1-30 inclusive. There are 24 fundamentally different graceful labelings (i.e., graceful labelings that are distinct modulo subtractive complementation and the symmetries of the graph), giving a total of 5760 graceful labelings in all (B. Dobbelaere, pers. comm., Oct. 2, 2020). The computation by Ashkok Kumar Chandra that found 5 fundamentally different solutions, as reported by Gardner (1983, pp. 163-164), therefore seems to be in error.
There are two minimal integral drawings of the icosahedral graph, illustrated above, both with maximum edge length of 8 (Harborth and Möller 1994).
The minimal planar integral drawing of the icosahedral graph has maximum edge length of 159 (Harborth et al. 1987).
The skeletons of the great dodecahedron, great icosahedron, and small stellated dodecahedron are all isomorphic to the icosahedral graph.
Removing any edge from the icosahedral graph gives the Tilley graph.
The chromatic polynomial of the icosahedral graph is
 |
and the chromatic number is 4.
Its graph spectrum is 
(Buekenhout and Parker 1998;
Cvetkovic et al. 1998, p. 310). Its automorphism
group is of group order 
(Buekenhout and Parker 1998).
The plots above show the adjacency matrix, incidence matrix, and graph distance matrix for the icosahedral graph.
The adjacency matrix for the icosahedral graph with 
appended, where 
is a unit matrix and 
is an identity matrix,
is a generator for the Golay code.
The following table summarizes properties of the icosahedral graph.
| property | value |
| automorphism group order | 120 |
| characteristic polynomial |  |
| chromatic number | 4 |
| claw-free | yes |
| clique number | 3 |
| determined by spectrum | ? |
| diameter | 3 |
| distance-regular graph | yes |
| dual graph name | dodecahedral graph |
| edge chromatic number | 5 |
| edge connectivity | 5 |
| edge count | 30 |
| Eulerian | no |
| girth | 3 |
| Hamiltonian | yes |
| Hamiltonian cycle count | 2560 |
| Hamiltonian path count | ? |
| integral graph | no |
| independence number | 3 |
| line graph | no |
| perfect matching graph | no |
| planar | yes |
| polyhedral graph | yes |
| polyhedron embedding names | great dodecahedron, great icosahedron, icosahedron, Jessen's orthogonal icosahedron, small stellated dodecahedron |
| radius | 3 |
| regular | yes |
| spectrum |  |
| square-free | no |
| traceable | yes |
| triangle-free | no |
| vertex connectivity | 5 |
| vertex count | 12 |
| weakly regular parameters |  |
."
Disc. Math. 186, 69-94, 1998.Cvetković, D. M.;
Doob, M.; and Sachs, H.