Cliques in graphs constructed from Strongly Orthogonal Subsets in exceptional root systems
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| Format: | Preprint |
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2026
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| author | Browne, Patrick J. Catháin, Pádraig Ó |
| author_facet | Browne, Patrick J. Catháin, Pádraig Ó |
| contents | Given a root system $R$, two roots are said to be \emph{strongly orthogonal} if neither their sum nor difference is a root. Gashi defined a family of graphs with vertices labelled by sums of $k$-element strongly orthogonal subsets of roots, and edges connect vertices whose difference is also a vertex. Gashi and the current authors established Erdős--Ko--Rado type results for graphs developed from Type $A$ root systems.
In this paper, we study graphs developed from the exceptional root systems $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$. We compute graph-theoretic invariants including regularity, connectivity, and clique numbers, and analyze clique structures with respect to sunflower properties. The automorphism group contains the Weyl group; we use these symmetries to obtain complete counts of maximum cliques and maximum sunflowers. Unlike type $A$, where all maximal cliques are sunflowers for large rank, sunflower cliques comprise at most 11\% of maximum cliques in the simply-laced exceptional types $E_6$, $E_7$, and $E_8$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cliques in graphs constructed from Strongly Orthogonal Subsets in exceptional root systems Browne, Patrick J. Catháin, Pádraig Ó Combinatorics 05C69, 05D05, 17B22 Given a root system $R$, two roots are said to be \emph{strongly orthogonal} if neither their sum nor difference is a root. Gashi defined a family of graphs with vertices labelled by sums of $k$-element strongly orthogonal subsets of roots, and edges connect vertices whose difference is also a vertex. Gashi and the current authors established Erdős--Ko--Rado type results for graphs developed from Type $A$ root systems. In this paper, we study graphs developed from the exceptional root systems $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$. We compute graph-theoretic invariants including regularity, connectivity, and clique numbers, and analyze clique structures with respect to sunflower properties. The automorphism group contains the Weyl group; we use these symmetries to obtain complete counts of maximum cliques and maximum sunflowers. Unlike type $A$, where all maximal cliques are sunflowers for large rank, sunflower cliques comprise at most 11\% of maximum cliques in the simply-laced exceptional types $E_6$, $E_7$, and $E_8$. |
| title | Cliques in graphs constructed from Strongly Orthogonal Subsets in exceptional root systems |
| topic | Combinatorics 05C69, 05D05, 17B22 |
| url | https://arxiv.org/abs/2604.02983 |