The Möbius-Kantor graph is the unique cubic symmetric graph on 16 nodes, illustrated above in a number of 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 Möbius-Kantor graph can be obtained as a subgraph of the Robertson graph by removing the three vertices and two edges illustrated above (pers. comm., E. Pegg, Jr., Oct. 27, 2025).
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.
The Möbius-Kantor graph is used in the construction of the Horton graphs. 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"].
