On an infinite family of integral and distance integral Cayley graphs with few distance eigenvalues

We construct the distance matrices of the Cayley graphs of the Pauli groups, recursively defined by using the notion of central product of groups, with respect to a suitable symmetric generating set. These Cayley graphs constitute an infinite sequence ${\left\{{\Gamma }_n\right\}}_{n\ge 1}$ of regular non distance-regular graphs of degree $3n+2$ with integral adjacency spectrum. We explicitly determine all the distance eigenvectors and eigenvalues, showing that each of these graphs has exactly six distinct distance eigenvalues, answering a question by Atik and Panigrahi. Moreover, all the distance eigenvalues have the remarkable property of being integers. We compute the diameter of each graph and the cardinality of all spheres centered at any vertex. We are also able to determine topological indices like the Wiener index, the Wiener polarity index, the Harary index, as well as the distance energy. We finally show that this sequence of Cayley graphs is a counterexample to two conjectures formulated by Aouchiche and Hansen.