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 ≤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 4×4 rook's graph), where one of them has quantum symmetry and the other one does not.