Extremal Cayley Digraphs of Finite Abelian Groups

Cayley graphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m_* (d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k element subset A of Γ such that diam(Cay(Γ, A)) ≤ d. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam(Z_m, A)) ≤ d. In this paper, we prove, among other results, that m_*(d, k) = m(d, k) for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley group is optimal with respect to its diameter and degree is always a cyclic group.