Gosset graph

Gosset graph
Gosset graph
	Gosset graph (321); (There are 3 rings of 18 vertices, and two vertices coincide in the center of this projection. 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
	Table of graphs and parameters

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.^[1]

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, or equivalently when their distance is $4{\sqrt {2}}$ .

An alternative construction is based on the 8-vertex complete graph K₈. The vertices of the Gosset graph can be identified with two copies of the set of edges of K₈. Two vertices of the Gosset graph that come from different copies are adjacent if they correspond to disjoint edges of K₈; two vertices that come from the same copy 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 K₈, two vertices of the Gosset graph are at distance three if and only if they correspond to different copies of the same edge of K₈. The Gosset graph is distance-regular with diameter three.^[2]

The induced subgraph of the neighborhood of any vertex in the Gosset graph is isomorphic to the Schläfli graph.^[2]

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^[3]

(x-27)(x-9)^{7}(x+1)^{27}(x+3)^{21}.\,

Therefore, this graph is an integral graph.

References

^ Grishukhin, V. P. (2011), "Delone and Voronoĭ polytopes of the root lattice $E 7$ and the dual lattice $E 7 *$ ", Trudy Matematicheskogo Instituta Imeni V. A. Steklova, 275: 68–86, doi:10.1134/S0081543811080049, MR 2962971, S2CID 120405049.
^ ^a ^b Kabanov, V. V.; Makhnev, A. A.; Paduchikh, D. V. (2007), "Characterization of some distance-regular graphs by forbidden subgraphs", Doklady Akademii Nauk, 414 (5): 583–586, doi:10.1134/S1064562407030234, MR 2451915, S2CID 119529234.
^ Brouwer, A. E.; Riebeek, R. J. (1998), "The spectra of Coxeter graphs", Journal of Algebraic Combinatorics, 8 (1): 15–28, doi:10.1023/A:1008670825910, MR 1635551.

External links

Weisstein, Eric W. "Gosset Graph". MathWorld.

[1] Grishukhin, V. P. (2011), "Delone and Voronoĭ polytopes of the root lattice $E 7$ and the dual lattice $E 7 *$ ", Trudy Matematicheskogo Instituta Imeni V. A. Steklova, 275: 68–86, doi:10.1134/S0081543811080049, MR 2962971, S2CID 120405049.

[kmp-2] Kabanov, V. V.; Makhnev, A. A.; Paduchikh, D. V. (2007), "Characterization of some distance-regular graphs by forbidden subgraphs", Doklady Akademii Nauk, 414 (5): 583–586, doi:10.1134/S1064562407030234, MR 2451915, S2CID 119529234.

[3] Brouwer, A. E.; Riebeek, R. J. (1998), "The spectra of Coxeter graphs", Journal of Algebraic Combinatorics, 8 (1): 15–28, doi:10.1023/A:1008670825910, MR 1635551.

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