The Möbius-Kantor graph is the unique cubic symmetric graph on 16 nodes, illustrated above in several embeddings. Its unique
canonical LCF notation is ![[5,-5]^8](/images/equations/Moebius-KantorGraph/Inline1.svg)
. The Möbius-Kantor graph is the Levi
graph of the Möbius-Kantor configuration
and can be constructed as the graph expansion
of 
with steps 1 and 3, where 
is a path graph (Biggs 1993,
p. 119).
The Möbius-Kantor graph is isomorphic to the generalized Petersen graph 
,
the Knödel graph 
, and honeycomb
toroidal graph 
.
The graph spectrum of the Möbius-Kantor graph is 
.
The Heawood graph is one of two cubic graphs on 16 nodes with smallest possible graph crossing number of 4 (the other being the 8-crossed prism graph), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).
It is also a unit-distance graph (Gerbracht 2008), as illustrated above.
A certain construction involving the Möbius-Kantor graph gives an infinite number of connected vertex-transitive graphs that have no Hamilton decomposition (Bryant and Dean 2014).
The plots above show the adjacency, incidence, and graph distance matrices for the Möbius-Kantor graph.
The Möbius-Kantor graph is implemented in the Wolfram Language as GraphData["MoebiusKantorGraph"].
The following table summarizes a number of properties for the Möbius-Kantor graph.
| property | value |
| automorphism group order | 96 |
| characteristic polynomial |  |
| chromatic number | 2 |
| chromatic polynomial |  |
| claw-free | no |
| clique number | 2 |
| graph complement name | ? |
| cospectral graph names | ? |
| determined by spectrum | no |
| diameter | 4 |
| distance-regular graph | no |
| dual graph name | ? |
| edge chromatic number | 3 |
| edge connectivity | 3 |
| edge count | 24 |
| edge transitive | yes |
| Eulerian | no |
| girth | 6 |
| Hamiltonian | yes |
| Hamiltonian cycle count | 12 |
| Hamiltonian path count | 1440 |
| integral graph | no |
| independence number | 8 |
| line graph | ? |
| line graph name | ? |
| perfect matching graph | no |
| planar | no |
| polyhedral graph | no |
| radius | 4 |
| regular | yes |
| square-free | yes |
| symmetric | yes |
| traceable | yes |
| triangle-free | yes |
| vertex connectivity | 3 |
| vertex count | 16 |
| vertex transitive | yes |
| weakly regular parameters | (16,(3),(0),(0,1)) |
