The Quantum Cayley Graphs of the Kac-Paljutkin Quantum Group

We study the quantum Cayley graphs associated with the Kac-Paljutkin quantum group. We determine which projections give rise to undirected, irreflexive and connected quantum graphs and classify them up to isomorphism. Their spectra and chromatic numbers are computed. For the special case of the projection $p=I_2$ we also compute the associated quantum Cuntz-Krieger algebra.