Extremal Problems in the Construction of Distributed Loop Networks

Abstract

Let $G(N, A)$ be the Cayley digraph associated with ${\bf Z}/(N)$ and $A$, where $N$ is a positive integer and $A$ is a subset of $\{1, 2, ..., N - 1\}$. Let $N(d, k)$ be the maximum $N$ such that the diameter of $G(N, A)$ is less than or equal to $d$ for some $A = \{a_{1}, a_{2},\ldots, a_{k}\}$ with $1 = a_{1}, < a_{2} < \cdots < a_{k}. An exact formula for $N(d, 2)$ is given, and $N(d, k)$ is estimated for $k \geq 3$. These results provide new bounds for minimal diameter in the construction of loop networks. A relation between this problem and the postage stamp problem in additive number theory is established to enhance the study of these problems.