Clebsch graph

Folded 5-cube
Vertices	16
Edges	40
Radius	2
Diameter	2
Girth	4
Automorphisms	1920
Chromatic number	4
Chromatic index	5
Properties	Strongly regular Hamiltonian Triangle-free Cayley graph Vertex-transitive Edge-transitive Distance-transitive.
Table of graphs and parameters

In the mathematical field of graph theory, the Clebsch graph is an undirected graph with 16 vertices and 80 edges. It was thus named by Seidel (1968) because of the relation to the configuration of 16 lines on the quartic surface discovered by the German mathematician Alfred Clebsch. It is the halved 5-cube, regular of valency 10.

Its complement, the folded 5-cube, which is regular of valency 5 and has no triangles, is also known as the Greenwood–Gleason graph after the work of Robert M. Greenwood and Andrew M. Gleason (1955), who used it to evaluate the Ramsey number R(3,3,3) = 17.

Confusion

Some authors, including MathWorld, use the name ‘Clebsch graph’ for the complement of the Clebsch graph as defined by Seidel. Let us here use Γ for the (original) Clebsch graph, and Δ for its complement, in order to avoid saying many times ‘complement of’.

Construction

The graph Δ is isomorphic to the order-5 folded cube graph. It may be constructed by adding edges between opposite pairs of vertices in a 4-dimensional hypercube graph. (In an n-dimensional hypercube, a pair of vertices are opposite if the shortest path between them has n edges.) Alternatively, it can be formed from a 5-dimensional hypercube graph by identifying together (or contracting) every opposite pair of vertices.

Another construction, leading to the same graph, is to create a vertex for each element of the finite field GF(16), and connect two vertices by an edge whenever the difference between the corresponding two field elements is a perfect cube.

Properties

The graph Γ is a strongly regular graph of degree 10 with parameters $(v,k,\lambda ,\mu )=(16,10,6,6)$. Its complement Δ is strongly regular of degree 5 with parameters $(v,k,\lambda ,\mu )=(16,5,0,2)$..

The graph Δ is hamiltonian, non planar and non eulerian. It is also both 5-vertex-connected and 5-edge-connected.

The subgraph that is induced by the ten non-neighbors of any vertex in Δ forms an isomorphic copy of the Petersen graph.

The edges of the complete graph K₁₆ may be partitioned into three disjoint copies of the graph Δ. Because Δ is a triangle-free graph, this shows that there is a triangle-free three-coloring of the edges of K₁₆; that is, that the Ramsey number R(3,3,3) describing the minimum number of vertices in a complete graph without a triangle-free three-coloring is at least 17. Greenwood & Gleason (1955) used this construction as part of their proof that R(3,3,3) = 17.

The graph Δ is the Keller graph of dimension two, part of a family of graphs used to find tilings of high-dimensional Euclidean spaces by hypercubes no two of which meet face-to-face.

Algebraic properties

The characteristic polynomial of Δ is $(x+3)^{5}(x-1)^{10}(x-5)$. Therefore Δ is an integral graph: its spectrum consists entirely of integers. The graph Δ is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

The Clebsch graph is a Cayley graph with an automorphism group of order 1920, isomorphic to the Coxeter group $D_{5}$. As a Cayley graph, its automorphism group acts transitively on its vertices, making it vertex transitive. In fact, both Γ and Δ are arc transitive, hence edge transitive and distance transitive.

Gallery

• 
  The graph Δ is Hamiltonian.
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  The achromatic number of the graph Δ is 8.
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  The chromatic number of the graph Δ is 4.
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  The chromatic index of the graph Δ is 5.
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  Construction of the graph Δ from a hypercube graph.