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.
|automorphism group order||96|
|graph complement name||?|
|cospectral graph names||?|
|determined by spectrum||no|
|dual graph name||?|
|edge chromatic number||3|
|Hamiltonian cycle count||12|
|Hamiltonian path count||1440|
|line graph name||?|
|perfect matching graph||no|
|weakly regular parameters||(16,(3),(0),(0,1))|