Large graphs of diameter two and given degree

Let r(d, 2), C(d, 2), and AC(d, 2) be the largest order of a regular graph, a Cayley graph, and a Cayley graph of an Abelian group, respectively, of diameter 2 and degree d. The best currently known lower bounds on these parameters are r(d, 2) ≥ d2 − d + 1 for d − 1 an odd prime power (with a similar result for powers of two), C(d, 2) ≥ (d+ 1)2/2 for degrees d = 2q − 1 where q is an odd prime power, and AC(d, 2) ≥ (3/8)(d2 − 4) where d = 4q − 2 for an odd prime power q. Using a number theory result on distribution of primes we prove, for all sufficiently large d, lower bounds on r(d, 2), C(d, 2), and AC(d, 2) of the form c · d2 − O(d1.525) for c = 1, 1/2, and 3/8, respectively. We also prove results of a similar flavour for vertextransitive graphs and Cayley graphs of cyclic groups.