On the quantum symmetry of distance-transitive graphs

Abstract

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs $H(n,3)$, the Johnson graphs $J(n,2)$ and the Kneser graphs $K(n,2)$ do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order $\le 20$ has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of graphs with the same intersection array (the Shrikhande graph and the $4\times 4$ rook's graph), where one of them has quantum symmetry and the other one does not.

arXiv comment43 pages

References in corpus (6)

• Quantum Symmetry Groups of Finite Spaces
• The hyperoctahedral quantum group
• The Morita theory of quantum graph isomorphisms
• Quantum automorphisms of folded cube graphs
• The Petersen graph has no quantum symmetry
• Study of quantum symmetries for vertex-transitive graphs using intertwiner spaces

Cited by in corpus (5)

• Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups
• Quantum symmetries of Cayley graphs of abelian groups
• Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach
• Solution group representations as quantum symmetries of graphs
• Uniformly vertex-transitive graphs