The Möbius-Kantor graph is the unique cubic symmetric graph on 16 nodes, illustrated above
in a number of drawings. 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 is a Cayley graph for exactly four nonisomorphic groups
of group order 16. They are the Pauli
group 
,
the semidihedral group 
, the semidirect product ![C_8×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]_5C_2](/images/equations/Moebius-KantorGraph/Inline9.svg)
,
and the dihedral group 
(cf. the three groups listed by Knill 2026). In ![C_8×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]_5C_2](/images/equations/Moebius-KantorGraph/Inline11.svg)
,
the nonidentity element of 
maps a generator 
of 
to 
. The automorphism
group of the Möbius-Kantor graph has group order
96 and is isomorphic to ![GL(2,3)×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]C_2](/images/equations/Moebius-KantorGraph/Inline16.svg)
(Knill 2026).
It is also a spanning subgraph of the tesseract graph 
,
obtained by deleting a perfect matching of eight
edges (Coxeter 1950, Knill 2026). It has a cellular
embedding on the torus with eight hexagonal graph
faces (Knill 2026).
The Möbius-Kantor graph is toroidal, as illustrated above. The left-hand drawing shows a graph drawing in a fundamental region whose paired boundary sides are identified to form a torus. The right-hand drawing shows a finite patch of the corresponding periodic lift, illustrating how edges continue across these boundaries and where corresponding vertices in different regions represent the same vertex on the torus.
The Möbius-Kantor graph can be obtained as a subgraph of the Robertson graph by removing the three vertices and two edges illustrated above (E. Pegg, Jr., pers. comm., Oct. 27, 2025).
The graph spectrum of the Möbius-Kantor graph is 
.
The line graph of the Möbius-Kantor graph has graph crossing number 12.
The Möbius-Kantor 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. 2020).
It is also a unit-distance graph (Gerbracht 2008), as illustrated above in a number of unit-distance embeddings.
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 matrices, incidence matrices, and graph distance matrices for the Möbius-Kantor graph.
The Möbius-Kantor graph is implemented in the Wolfram Language as GraphData["MoebiusKantorGraph"].
