The Gosset graph, named after

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) 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, a ...

, is a specific regular graph (1-

skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...

of the 7-dimensional 3₂₁ polytope) 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. An alternative construction is based on the 8-vertex

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 ...

''K''₈. The vertices of the Gosset graph can be identified with two copies of the set of edges of ''K''₈. Two vertices of the Gosset graph that come from the same copy are adjacent if they correspond to disjoint edges of ''K''₈; two vertices that come from different copies are adjacent if they correspond to edges that share a single vertex.

Properties

In the vector representation of the Gosset graph, two vertices are at distance two when their 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). In the representation based on the edges of ''K''₈, two vertices of the Gosset graph are at distance three if and only if they correspond to different copies of the same edge of ''K''₈. The Gosset graph is distance-regular with diameter three.. The

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. Definit ...

of the neighborhood of any vertex in the Gosset graph is isomorphic to the Schläfli graph. The

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is th ...

of the Gosset graph is isomorphic to the

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 reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

''E''₇ and hence has order 2903040. The Gosset 3₂₁ polytope is a

semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...

. Therefore, the automorphism group of the Gosset graph, ''E''₇, acts transitively upon its vertices, making it a

vertex-transitive graph In the mathematical field of graph theory, a vertex-transitive graph is a graph in which, given any two vertices and of , there is some automorphism :f : G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive ...

. 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 ...

of the Gosset graph is. :

(x-27)(x-9)^7(x+1)^(x+3)^. \,

Therefore, this graph is an integral graph.

References

External links

* {{MathWorld, title=Gosset Graph, urlname=GossetGraph Individual graphs Regular graphs