# On the quantum symmetry of distance-transitive graphs

## TL;DR

This paper investigates quantum automorphism groups of distance-transitive graphs, establishing which graphs lack quantum symmetry, providing a comprehensive table for cubic graphs, and analyzing specific examples including the Hoffman-Singleton and Shrikhande graphs.

## Contribution

It identifies classes of distance-transitive graphs without quantum symmetry and offers a complete table for cubic graphs, advancing understanding of quantum symmetries in graph theory.

## Key findings

- Odd graphs, Hamming $H(n,3)$, Johnson $J(n,2)$, and Kneser $K(n,2)$ graphs lack quantum symmetry.
- The Hoffman-Singleton graph has no quantum symmetry.
- A pair of graphs with identical intersection arrays shows differing quantum symmetry properties.

## 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 $\leq 20$ 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.