The Möbius-Kantor graph is the unique cubic symmetric graph on 16 nodes, illustrated above in several embeddings. Its unique canonical LCF notation is . 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)) |