Extremal Cayley Digraphs of Finite Cyclic Groups
Xing‐De Jia · SIAM Journal on Discrete Mathematics · 1995
Let Cay$( m,A t)$ denote the Cayley digraph of $\mathbf{Z}_m $ generated by A, where $\mathbf{Z}_m $ is the cyclic group of residues modulo m. Let $r( m,A )$ denote the average distance of Cay$(m,A)$. For any $r \geq 1$ and $k \geq 1$ define $m*(r,k)$ as the largest positive integer m such that the average distance of the Cayley digraph Cay$(m,A)$ is at most r for some set A with k elements. In this paper, an asymptotic formula for $m*(r,2)$ is proved and a lower bound for $m*(r,k)$ is also obtained for $k \geq 3$. Applications to the construction of optimal distributed loop networks are discussed in this paper. A lower bound of the average order of subsets for asymptotic bases in number theory is proved using the main theorem of this paper.