Wagner graph
The Wagner graph
Named after	Klaus Wagner
Vertices	8
Edges	12
Radius	2
Diameter	2
Girth	4
Automorphisms	16 (D8)
Chromatic number	3
Chromatic index	3
Genus	1
Spectrum	${\displaystyle {\begin{pmatrix}3&1&{\sqrt {2}}-1&-1&-{\sqrt {2}}-1\\1&2&2&1&2\end{pmatrix}}}$
Properties	Cubic Hamiltonian Triangle-free Vertex-transitive Toroidal Apex
Notation	M8
Table of graphs and parameters

In the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges.^[1] It is the 8-vertex Möbius ladder graph.

Properties

As a Möbius ladder, the Wagner graph is nonplanar but has crossing number one, making it an apex graph. It can be embedded without crossings on a torus or projective plane, so it is also a toroidal graph. It has girth 4, diameter 2, radius 2, chromatic number 3, chromatic index 3 and is both 3-vertex-connected and 3-edge-connected.

The Wagner graph has 392 spanning trees; it and the complete bipartite graph K_3,3 have the most spanning trees among all cubic graphs with the same number of vertices.^[2]

The Wagner graph is a vertex-transitive graph but is not edge-transitive. Its full automorphism group is isomorphic to the dihedral group D₈ of order 16, the group of symmetries of an octagon, including both rotations and reflections.

The characteristic polynomial of the Wagner graph is

${\displaystyle (x-3)(x-1)^{2}(x+1)(x^{2}+2x-1)^{2}.}$

It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

The Wagner graph is triangle-free and has independence number three, providing one half of the proof that the Ramsey number R(3,4) (the least number n such that any n-vertex graph contains either a triangle or a four-vertex independent set) is 9.^[3]

Graph minors

Möbius ladders play an important role in the theory of graph minors. The earliest result of this type is a 1937 theorem of Klaus Wagner (part of a cluster of results known as Wagner's theorem) that graphs with no K₅ minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M₈.^[4] For this reason M₈ is called the Wagner graph.

The Wagner graph is also one of four minimal forbidden minors for the graphs of treewidth at most three (the other three being the complete graph K₅, the graph of the regular octahedron, and the graph of the pentagonal prism) and one of four minimal forbidden minors for the graphs of branchwidth at most three (the other three being K₅, the graph of the octahedron, and the cube graph).^[5][6]

Construction

The Wagner graph is a cubic Hamiltonian graph and can be defined by the LCF notation [4]⁸. It is an instance of an Andrásfai graph, a type of circulant graph in which the vertices can be arranged in a cycle and each vertex is connected to the other vertices whose positions differ by a number that is 1 (mod 3). It is also isomorphic to the circular clique K_8/3.

It can be drawn as a ladder graph with 4 rungs made cyclic on a topological Möbius strip.

Gallery

• 
  The chromatic number of the Wagner graph is 3.
• 
  The chromatic index of the Wagner graph is 3.
• 
  The Wagner graph drawn on the Möbius strip.

References

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