In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, the Coxeter graph is a 3-

regular graph In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree o ...

with 28 vertices and 42 edges. It is one of the 13 known cubic

distance-regular graph In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . ...

s. It is named after

Harold Scott MacDonald Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

Properties

The Coxeter graph has

chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices o ...

chromatic index In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue ...

3, radius 4, diameter 4 and

girth Girth may refer to: ;Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth ...

7. It is also a 3- vertex-connected graph and a 3- edge-connected graph. It has

book thickness In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a ''book'', a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie ...

3 and

queue number In the mathematical field of graph theory, the queue number of a graph is a graph invariant defined analogously to stack number (book thickness) using first-in first-out (queue) orderings in place of last-in first-out (stack) orderings. Defi ...

2. 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 .

Construction

The simplest construction of a Coxeter graph is from a

Fano plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines c ...

. Take the ⁷C₃ = 35 possible 3-combinations on 7 objects. Discard the 7 triplets that correspond to the lines of the Fano plane, leaving 28 triplets. Link two triplets if they are disjoint. The result is the Coxeter graph. (See

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

.) This construction exhibits the Coxeter graph as an

induced subgraph In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Defini ...

of the

Kneser graph In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs a ...

. The Coxeter graph may also be constructed from the smaller distance-regular

Heawood graph Heawood is a surname. Notable people with the surname include: *Jonathan Heawood, British journalist *Percy John Heawood (1861–1955), British mathematician **Heawood conjecture **Heawood graph **Heawood number In mathematics, the Heawood number ...

by constructing a vertex for each 6-cycle in the Heawood graph and an edge for each disjoint pair of 6-cycles. The Coxeter graph may be derived from the Hoffman-Singleton graph. Take any vertex ''v'' in the Hoffman-Singleton graph. There is an independent set of size 15 that includes ''v''. Delete the 7 neighbors of ''v'', and the whole independent set including ''v'', leaving behind the Coxeter graph.

Algebraic properties

The automorphism group of the Coxeter graph is a group of order 336. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Coxeter graph is a

symmetric graph In the mathematical field of graph theory, a graph is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices and of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 and f(v_1) = v_2. In oth ...

. 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. The Coxeter graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...

. As a finite connected vertex-transitive graph that contains no

Hamiltonian cycle In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...

, 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 In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...

''K''₂, the

Petersen graph In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is n ...

, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.Royle, G
"Cubic Symmetric Graphs (The Foster Census)."
The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...

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.

Gallery

Layouts

These are different representations of the Coxeter graph, using the same vertex labels. There are four colors, and seven vertices of each color.
Each red, green or blue vertex is connected with two vertices of the same color (thin edges forming 7-cycles) and to one white vertex (thick edges).

Properties

References

* Coxeter, H. S. M. "My Graph." Proc. London Math. Soc. 46, 117-136, 1983. {{Authority control Individual graphs Regular graphs