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 D ₁(N) = min {D(N; 1, s)|1 < s < N }. Xu and Aguil$\acute{o}$ et al. gave some infinite families of 0-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.