Abelian Cayley Graphs of Given Degree and Diameter 2 and 3

Let CCd,k be the largest possible number of vertices in a cyclic Cayley graph of degree d and diameter k, and let ACd,k be the largest order in an Abelian Cayley graph for given d and k. We show that $${CC_{d,2} \geq \frac{13}{36} (d + 2)(d -4)}$$CCd,2≥1336(d+2)(d-4) for any d= 6p−2 where p is a prime such that $${p \neq 13}$$p≠13 , $${p \not\equiv 1}$$p≢1 (mod 13), and $${AC_{d,3} \geq \frac{9}{128} (d + 3)^2(d - 5)}$$ACd,3≥9128(d+3)2(d-5) for d = 8q−3 where q is a prime power.