Noncommutative properties of 0-hyperbolic graphs
Amaury Freslon, Paul Meunier, Pegah Pournajafi
TL;DR
The paper investigates noncommutative properties of graphs through quantum symmetries, proving that $0$-hyperbolicity is preserved under quantum isomorphism and identifying block graphs as the fundamental class underpinning this phenomenon. It develops a robust decomposition toolkit based on partitioned graphs and the $ ext{Ψ}$-operation to analyze quantum automorphism groups, showing that block graphs have quantum automorphism groups in $ ext{Jor}^+(oldsymbol{ ootnotesize oldsymbol{ extbf{C}})$ and that block graphs form a superrigid tractable family. The results extend to rooted block graphs and to block-cographs, preserving the same noncommutative-symmetry profile across these families. This work links classical block structure with noncommutative symmetries, providing a structural, scalable framework for understanding quantum automorphisms in large graph classes and suggesting a path toward classifying hyperbolicity under quantum equivalence.
Abstract
We study several noncommutative properties of 0-hyperbolic graphs. In particular, we prove that 0-hyperbolicity is preserved under quantum isomorphism. We also compute the quantum automorphism groups of 0-hyperbolic graphs and characterise the ones with quantum symmetry.