Quantum graphs and spin models

We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from deforming graphs using higher-idempotent splittings of quantum isomorphisms for which we prove that the relevant algebraic, combinatorial, and topological properties of the original graphs are preserved along with the quantum automorphism group. We also obtain exotic examples of highly regular quantum graphs using the quantum Fourier transform and a method of iterated convolution. Our examples include quantum versions of the strongly regular $9$-Paley, $16$-Clebsch and the Higman-Sims graphs, yielding new models for their regularity parameters. As applications, we construct a compact quantum group that is monoidally equivalent to $SO_q(5)$ at the square of the golden ratio, whose dual is infinite with property (T), and exhibit a highly-regular quantum graph with no classical analogue. Finally, we introduce quantum spin models, construct explicit examples and make contact with quantum Hadamard matrices.