Title: Quantum Isomorphic Graphs from Association Schemes
Abstract: This talk discusses joint work with Ada Chan (York University) on the quantum isomorphism game. In rough terms, we showed that two corresponding graphs coming from association schemes with the same parameters are quantum isomorphic if the association schemes are both triply regular and dually triply regular (with the “same” parameters). This was the first construction of more than two graphs that are pairwise quantum isomorphic yet not isomorphic. Our examples are Hadamard graphs and, in fact, the number of non-isomorphic Hadamard graphs on 4n vertices grows exponentially with n (for some growing sequence {n} of integers). The main theorem is built of three key components. Most importantly, we use a result of Laura Mančinska and David Roberson establishing the fact that graphs G and H are quantum isomorphic if and only if they admit equally many homomorphism from any planar graph. We also use a classical result of Epifanov that reduces any planar graph to a copy of K_1 via extended series-parallel reductions and Delta-Wye transformations. The third tool is the theory of scaffolds that I built based on ideas of Neumaier and Terwilliger to simplify counting and similar computations in association schemes.