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

In this article, we investigate the bound of the valency of the Cayley graphs of the generalized quaternion groups which guarantees to be Ramanujan. As is the cases of the cyclic and dihedral groups in our previous studies, we show that the determination of the bound in a special setting is related to the classical Hardy-Littlewood conjecture for primes represented by a quadratic polynomial.