Cayley graphs of given degree and diameter for cyclic, Abelian, and metacyclic groups

Let C C ( d , 2 ) and A C ( d , 2 ) be the largest order of a Cayley graph of a cyclic and an Abelian group, respectively, of diameter 2 and a given degree d . There is an obvious upper bound of the form C C ( d , 2 ) ≤ A C ( d , 2 ) ≤ d 2 / 2 + d + 1 . We prove a number of lower bounds on both quantities for certain infinite sequences of degrees d related to primes and prime powers, the best being C C ( d , 2 ) ≥ ( 9 / 25 ) ( d + 3 ) ( d − 2 ) and A C ( d , 2 ) ≥ ( 3 / 8 ) ( d 2 − 4 ) . We also offer a result for Cayley graphs of metacyclic groups for general degree and diameter.