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.