Gosset graph

Gosset graph

Gosset graph (321)
(Two vertices coincide in the center of this graph. Edges also coincide with this projection.)
Named after Thorold Gosset
Vertices 56
Edges 756
Radius 3
Diameter 3
Girth 3
Automorphisms 2903040
Properties Distance-regular graph
Integral
Vertex-transitive

The Gosset graph, named after Thorold Gosset, is a specific regular graph (1-skeleton 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.

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 with diameter three.

The automorphism group of the Gosset graph is isomorphic to the Coxeter group 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 upon its vertices, making it a vertex-transitive graph.

The characteristic polynomial of the Gosset graph is

(x-27)(x-9)^7(x%2B1)^{27}(x%2B3)^{21}. \,

Therefore this graph is an integral graph.

References