Infinite families of optimal double-loop networks

Abstract

A double-loop network(DLN) G(N; r, s) is a digraph with the vertex set V = {0,1,..., N - 1} and the edge set E = {v -→ v + r(mod N) and v → v + s( mod N)|v ∈V }. Let D(N; r, s) be the diameter of G, D(N) = min{D(N;r,s)|1 ≤ r < s < N and gcd(N;r,s) = 1} and D1(N) = min{D(N;1,s)|1 < s < N}. Xu and Aguiló et al. gave some infinite families of 0-tight non-unit step (nus) integers with D1(N) - D(N) > 1. In this paper, an approach is proposed for finding infinite families of k-tight(k ≥ 0) optimal double-loop networks G(N;r,s), and two infinite families of k-tight optimal double-loop networks G(N; r, s) are presented. We also derive one infinite family of 1tight nus integers with D1(N) - D(N) ≥ 1 and one infinite family of 1-tight nus integers with D 1(N) - D(N) ≥ 2. As a consequence of these works, some results by Xu are improved. © Springer-Verlag Berlin Heidelberg 2007.