Möbius-Kantor Graph


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. The Möbius-Kantor graph is the Levi graph of the Möbius-Kantor configuration and can be constructed as the graph expansion of 8P_2 with steps 1 and 3, where P_2 is a path graph (Biggs 1993, p. 119).

The Möbius-Kantor graph is isomorphic to the generalized Petersen graph GP(8,3), the Knödel graph W_(3,16), and honeycomb toroidal graph HTG(1,16,5).

The graph spectrum of the Möbius-Kantor graph is (-3)^1(-sqrt(3))^4(-1)^31^3(sqrt(3))^43^1.

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 order96
characteristic polynomial(x-3)(x-1)^3(x+1)^3(x+3)(x^2-3)^4
chromatic number2
chromatic polynomial(x-1)x(x^(14)-23x^(13)+253x^(12)-1771x^(11)+8855x^(10)-33625x^9+100515x^8-241471x^7+470570x^6-743126x^5+938926x^4-922082x^3+665670x^2-315822x+74037)
clique number2
graph complement name?
cospectral graph names?
determined by spectrumno
distance-regular graphno
dual graph name?
edge chromatic number3
edge connectivity3
edge count24
edge transitiveyes
Hamiltonian cycle count12
Hamiltonian path count1440
integral graphno
independence number8
line graph?
line graph name?
perfect matching graphno
polyhedral graphno
vertex connectivity3
vertex count16
vertex transitiveyes
weakly regular parameters(16,(3),(0),(0,1))

See also

Cubic Symmetric Graph, Honeycomb Toroidal Graph, Möbius-Kantor Configuration, Smallest Cubic Crossing Number Graph

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Brouwer, A. E. "Möbius-Kantor Graph.", D. and Dean, M. "Vertex-Transitive Graphs that have no Hamilton Decomposition." 25 Aug 2014., K.; Haythorpe, M.; Newcombe, A.; and Pegg, E. Jr. "There Are No Cubic Graphs on 26 Vertices with Crossing Number 10 or 11." Preprint. 2019.Coxeter, H. S. M. "Self-Dual Configurations and Regular Graphs." Bull. Amer. Math. Soc. 56, 413-455, 1950.Gerbracht, E. H.-A. "On the Unit Distance Embeddability of Connected Cubic Symmetric Graphs." Kolloquium über Kombinatorik. Magdeburg, Germany. Nov. 15, 2008.Pegg, E. Jr. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 161-170, 2009.

Cite this as:

Weisstein, Eric W. "Möbius-Kantor Graph." From MathWorld--A Wolfram Web Resource.

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