Download Wolfram Notebook

The Wagner graph is a name sometimes given to the 4-Möbius ladder (Bondy and Murty 2008, pp. 275-276). The association arises through the theorem of Wagner (1937) that graphs having no minor can be constructed using clique-sum operations to combine planar graphs and this graph. It is illustrated above in a number of drawings.

The three-dimensional locally twisted cube graph is isomorphic to the Wagner graph. Thus the Wagner graph is the first non-bipartite member of this recursively defined family.

The Wagner graph has the most spanning trees among the six 8-vertex cubic graphs, namely 392.

It is a toroidal graph, as illustrated above. The left-hand drawing shows an embedding 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 Wagner graph is implemented in the Wolfram Language as GraphData["WagnerGraph"].

See also

Möbius Ladder

Explore with Wolfram|Alpha

References

Bondy, J. A. and Murty, U. S. R. Graph Theory. Berlin, Germany: Springer-Verlag, pp. 275-276, 2008.House of Graphs. "Wagner Graph ." https://houseofgraphs.org/graphs/640.Wagner, K. "Über eine Eigenschaft der ebenen Komplexe." Math. Ann. 114, 570-590, 1937.

Referenced on Wolfram|Alpha

Wagner Graph

Cite this as:

Weisstein, Eric W. "Wagner Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WagnerGraph.html

Subject classifications