Infinite Families of Optimal Double-Loop Networks

摘要

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 D_1(N) = min{D{N; 1, s)|1 ＜ s ＜ N}. Xu and Aguild et al. gave some infinite families of O-tight non-unit step(nus) integers with D_1(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 1-tight nus integers with D_1(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.