Quantum isomorphic strongly regular graphs from the $E_8$ root system

TL;DR

This paper presents the first known example of a pair of quantum isomorphic, non-isomorphic strongly regular graphs with identical parameters, derived from the $E_8$ root system, expanding understanding of quantum graph symmetries.

Contribution

It introduces the first example of quantum isomorphic, non-isomorphic strongly regular graphs, and demonstrates how to generate more using Godsil-McKay switching.

Findings

Identified a pair of quantum isomorphic, non-isomorphic strongly regular graphs with parameters (120, 63, 30, 36).

Used the $E_8$ root system to construct these graphs.

Applied Godsil-McKay switching to produce additional such graph pairs.

Abstract

In this article, we give a first example of a pair of quantum isomorphic, non-isomorphic strongly regular graphs, that is, non-isomorphic strongly regular graphs having the same homomorphism counts from all planar graphs. The pair consists of the orthogonality graph of the $120$ lines spanned by the $E_8$ root system and a rank $4$ graph whose complement was first discovered by Brouwer, Ivanov and Klin. Both graphs are strongly regular with parameters $(120, 63, 30, 36)$. Using Godsil-McKay switching, we obtain more quantum isomorphic, non-isomorphic strongly regular graphs with the same parameters.