Cliques in graphs constructed from Strongly Orthogonal Subsets in exceptional root systems

Abstract

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 Erd\H{o}s--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 $E_8$.