TOPICS
Search

Möbius-Kantor Graph

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
MoebiusKantorGraphEmbeddings

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. 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).

MoebiusKantorFromRobertsonGraph

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 (-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. 2020).

MoebiusKantorGraphUnitDistance

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).

Moebius-KantorGraphMatrices

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"].

See also

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

Explore with Wolfram|Alpha

References

Brouwer, A. E. "Möbius-Kantor Graph." https://aeb.win.tue.nl/drg/graphs/MoebiusKantor.html.Bryant, D. and Dean, M. "Vertex-Transitive Graphs that have no Hamilton Decomposition." 25 Aug 2014. https://arxiv.org/abs/1408.5211.Clancy, K.; Haythorpe, M.; Newcombe, A.; and Pegg, E. Jr. "There Are No Cubic Graphs on 26 Vertices with Crossing Number 10 or 11." Graphs Combin. 36, 1713-1721, 2020. https://doi.org/10.1007/s00373-020-02204-6.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.House of Graphs. "Moebius Kantor Graph." https://houseofgraphs.org/graphs/1229.Pegg, E. Jr. and Exoo, G. "Crossing Number Graphs." Mathematica J. 11, 161-170, 2009. https://doi.org/10.3888/tmj.11.2-2.

Referenced on Wolfram|Alpha

Möbius-Kantor Graph

Cite this as:

Weisstein, Eric W. "Möbius-Kantor Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Moebius-KantorGraph.html

Subject classifications