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

TL;DR

This paper investigates the structure of graphs derived from exceptional root systems, focusing on invariants like clique numbers and sunflower properties, and reveals differences from type A systems.

Contribution

It extends the study of graphs from strongly orthogonal roots to exceptional root systems, providing new invariants and detailed clique analyses.

Findings

Computed graph invariants such as regularity, connectivity, and clique numbers.

Analyzed the structure of maximum cliques and sunflowers in exceptional types.

Found that sunflower cliques are a small fraction of maximum cliques in certain types.

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 $E_8$.