# Coxeter graph

Coxeter graph
The Coxeter graph
Vertices 28
Edges 42
Diameter 4
Girth 7
Automorphisms 336 (PGL2(7))
Chromatic number 3
Chromatic index 3
Properties Symmetric
Distance-regular
Distance-transitive
Cubic
Hypohamiltonian

In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges.[1] All the cubic distance-regular graphs are known.[2] The Coxeter graph is one of the 13 such graphs.

## Properties

The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth 7. It is also a 3-vertex-connected graph and a 3-edge-connected graph.

The Coxeter graph is hypohamiltonian : it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has rectilinear crossing number 11, and is the smallest cubic graph with that crossing number currently known, but an 11-crossing, 26-vertex graph may exist (sequence A110507 in OEIS).

The Coxeter graph may be constructed from the smaller distance-regular Heawood graph by constructing a vertex for each 6-cycle in the Heawood graph and an edge for each disjoint pair of 6-cycles.[3]

## Algebraic properties

The automorphism group of the Coxeter graph is a group of order 336.[4] It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Coxeter graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Coxeter graph, referenced as F28A, is the only cubic symmetric graph on 28 vertices.[5]

The Coxeter graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.[6]

As a finite connected vertex-transitive graph that contains no Hamiltonian cycle, the Coxeter graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for an Hamiltonian path and is verified by the Coxeter graph.

Only five examples of vertex-transitive graph with no Hamiltonian cycles are known : the complete graph K2, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.[7]

The characteristic polynomial of the Coxeter graph is $(x-3) (x-2)^8 (x+1)^7 (x^2+2 x-1)^6$. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

## References

1. ^
2. ^ Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
3. ^ Dejter, Italo J. (2011), "From the Coxeter graph to the Klein graph", Journal of Graph Theory, arXiv:1002.1960, doi:10.1002/jgt.20597.
4. ^ Royle, G. F028A data
5. ^ Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.
6. ^ E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189-202, 2003
7. ^