Abstract
AbstractLet $$\varGamma $$
Γ
be the graph on the roots of the $$E_8$$
E
8
root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$
Γ
c
be the subgraph of $$\varGamma $$
Γ
consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$
Γ
that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$
Γ
c
for some color set c, or whose vertices are the vertices of a face of the $$E_8$$
E
8
root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$
Γ
if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$
Γ
, in terms of the restrictions of f to certain special subgraphs of K of size at most 7.
Funder
Nederlandse Organisatie voor Wetenschappelijk Onderzoek
Max Planck Institute for Mathematics in the Sciences
Publisher
Springer Science and Business Media LLC
Subject
Discrete Mathematics and Combinatorics,Theoretical Computer Science
Reference12 articles.
1. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997). https://doi.org/10.1006/jsco.1996.0125. (Computational algebra and number theory, London, 1993)
2. Bourbaki, N.: Groupes et algèbres de Lie, Chapitres 4, 5 et 6. Éléments de mathématique. Masson (1981)
3. Coxeter, H.S.M.: The polytopes with regular-prismatic vertex figures. Philos. Trans. R. Soc. Lond. 229, 329–425 (1930)
4. Coxeter, H.S.M.: Regular Polytopes. Methuen & co. LTD., London (1948)
5. Cvetković, D., Rowlinson, P., Simić, S.: Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue -2, London Mathematical Society Lecture Note Series, vol. 314. Cambridge University Press, Cambridge (2004)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献