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

TL;DR

This paper explores the conditions under which Cayley graphs of generalized quaternion groups are Ramanujan, revealing a connection to the Hardy-Littlewood conjecture about primes represented by quadratic polynomials.

Contribution

It establishes a link between the Ramanujan property of Cayley graphs of generalized quaternion groups and the Hardy-Littlewood conjecture, extending previous work on cyclic and dihedral groups.

Findings

Derived bounds for valency ensuring Ramanujan property

Connected graph properties to Hardy-Littlewood conjecture

Extended understanding of Cayley graphs of quaternion groups

Abstract

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.