Quantum isomorphism of graphs from association schemes

TL;DR

This paper demonstrates that graphs derived from certain association schemes, including Hadamard graphs, are quantum isomorphic, using a combination of recent theoretical results, generalized partition functions, and graph reduction techniques.

Contribution

It provides a general method for establishing quantum isomorphism of graphs from association schemes, extending previous results to a broader class of graphs.

Findings

Hadamard graphs are quantum isomorphic when on the same number of vertices

Graphs from certain association schemes are quantum isomorphic

The method combines homomorphism counting, scaffolds, and graph reductions

Abstract

We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result of Man\v{c}inska and Roberson shows that graphs $G$ and $H$ are quantum isomorphic if and only if, for any planar graph $F$, the number of graph homomorphisms from $F$ to $G$ is equal to the number of graph homomorphisms from $F$ to $H$. A generalization of partition functions called "scaffolds" affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research.