Ramanujan Cayley Graphs of the Generalized Quaternion Groups and the Hardy–Littlewood Conjecture
Ramanujan graphs are graphs which have many connections with various mathematical fields including an application to the cryptography. In this article, as a continuous work of our research on Ramanujan graphs, 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, 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.
KeywordsRamanujan graphs Generalized quaternion groups Hardy–Littlewood conjecture
The author thanks the referee for helpful comments on the manuscript. Moreover, the author also would like to thank Miki Hirano and Kohei Katata for variable discussion.
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