Ramanujan Cayley graphs of the generalized quaternion groups and the Hardy-Littlewood conjecture

摘要

In this article, we investigate the bound of the valency of the Cayley graphsof the generalized quaternion groups which guarantees to be Ramanujan. As isthe cases of the cyclic and dihedral groups in our previous studies, we showthat the determination of the bound in a special setting is related to theclassical Hardy-Littlewood conjecture for primes represented by a quadraticpolynomial.