The Gosset graph, named after Thorold Gosset

Thorold Gosset

Thorold Gosset was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.According to H. S. M...

, is a specific regular graph (1-skeleton

N-skeleton

In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions m ≤ n...

 of the 7-dimensional 3₂₁ polytope

Gosset 3 21 polytope

In 7-dimensional geometry, the 321 polytope is a uniform 6-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper...

) with 56 vertices and valency 27.

Construction

The Gosset graph can be explicitly constructed as follows: the 56 vertices are the vectors in R⁸, obtained by permuting the coordinates and possibly taking the opposite of the vector (3, 3, −1, −1, −1, −1, −1, −1). Two such vectors are adjacent when their inner product is 8.

Properties

In the above representation, two vertices are at distance two when ther inner product is −8 and at distance three when their inner product is −24 (which is only possible if the vectors are each other's opposite). The Gosset graph is distance-regular

Distance-regular graph

In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w at distance i the number of vertices adjacent to w and at distance j from v is the same. Every distance-transitive graph is distance-regular...

 with diameter three.

The automorphism group

Graph automorphism

In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity....

 of the Gosset graph is isomorphic to the Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

 E₇ and hence has order 2903040. The Gosset 3₂₁ polytope is a semiregular polytope. Therefore the automorphism group of the Gosset graph, E₇, acts transitively

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 upon its vertices, making it a vertex-transitive graph

Vertex-transitive graph

In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphismf:V \rightarrow V\ such thatf = v_2.\...

.

The characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 of the Gosset graph is

Therefore this graph is an integral graph

Integral graph

In the mathematical field of graph theory, an integral graph is a graph whose spectrum consists entirely of integers. In other words, a graphs is an integral graph if all the eigenvalues of its characteristic polynomial are integers....

.