Large bipartite Cayley graphs of given degree and diameter

Let B C d , k be the largest possible number of vertices in a bipartite Cayley graph of degree d and diameter k . We show that B C d , k ≥ 2 ( k − 1 ) ( ( d − 4 ) / 3 ) k − 1 for any d ≥ 6 and any even k ≥ 4 , and B C d , k ≥ ( k − 1 ) ( ( d − 2 ) / 3 ) k − 1 for d ≥ 6 and k ≥ 7 such that k ≡ 3 (mod 4 ).