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Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree$d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartiteCayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for everyodd $k$ we construct Cayley digraphs of order$2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ and diameter at most $k$, where$k\ge 3$, and bipartite Cayley digraphs of order$2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ and diameter at most $k$,where $k\ge 5$. These constructions yield the bounds $C_{d,k} \ge2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ for odd $k\ge 3$ and $d\ge\frac{3^{k}}{2k}+1$, and $BC_{d,k} \ge2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ for odd $k\ge 5$ and $d\ge\frac{3^{k-1}}{k-1}+1$. Our constructions give the best currently known boundson the orders of large Cayley digraphs and bipartite Cayley digraphs of givendegree and odd diameter $k\ge 5$. In our proofs we use new techniques based onproperties of group automorphisms of direct products of abelian groups.

Large Cayley digraphs and bipartite Cayley digraphs of odd diameters