Large Cayley digraphs and bipartite Cayley digraphs of odd diameters

Let Cd,k be the largest number of vertices in a Cayley digraph of degree d and diameter k, and let BCd,k be the largest order of a bipartite Cayley digraph for given d and k. For every degree d ≥ 2 and for every odd k we construct Cayley digraphs of order 2k ( ⌊d2⌋ )k and diameter at most k, where k ≥ 3, and bipartite Cayley digraphs of order 2(k − 1) ( ⌊d2⌋ )k−1 and diameter at most k, where k ≥ 5. These constructions yield the bounds Cd,k ≥ 2k ( ⌊d2⌋ )k for odd k ≥ 3 and d ≥ 3 k 2k + 1, and BCd,k ≥ 2(k − 1) ( ⌊d2⌋ )k−1 for odd k ≥ 5 and d ≥ 3 k−1 k−1 + 1. Our constructions give the best currently known bounds on the orders of large Cayley digraphs and bipartite Cayley digraphs of given degree and odd diameter k ≥ 5. In our proofs we use new techniques based on properties of group automorphisms of direct products of abelian groups.