Cayley graphs of given degree and diameters 3, 4 and 5

Let C d , k be the largest number of vertices in a Cayley graph of degree d and diameter k . We show that C d , 3 ≥ 3 16 ( d − 3 ) 3 and C d , 5 ≥ 25 ( d − 7 4 ) 5 for any d ≥ 8 , and C d , 4 ≥ 32 ( d − 8 5 ) 4 for any d ≥ 10 . For sufficiently large d our graphs are the largest known Cayley graphs of degree d and diameters 3, 4 and 5.