On infinite families of optimal double-loopnetworks with non-unit steps

摘要

Double-loop networks have been widely studied as architecture for local area networks. A double-loop network G(N; s1, s2) is a digraph with N vertices 0, 1,..., N – 1 and 2N edges of two types: si-edge: i→i+si( mod N); i = 0,1, ... , N – 1.s2-edge: i →i+ s2( mod N); i = 0,1, ... ,N – 1.for some fixed steps 1 ≤ s1 < s2 < N with gcd(N, s1, s2) = 1. Let D(N; s1, s2) be the diameter of G and let us define D(N) = min{D(N; s1, a2)1≤ S1 < s2 < N and gcd(N, s1, s2) = 1}, and D_1(N) = min{D(N; 1, s)11 < s < N}. If N is a positive integer and D(N) < D_1(N), then N is called a non-unit step integer or a nus integer. Xu and Aguilo et al. gave some infinite families of 0-tight nus integers with D_1 (N) – D(N) ≥ 1.In this work, we give a method for finding infinite families of nus integers. As application examples, we give one infinite family of 0-tight nus integers with D_1 (N) – D(N) ≥ 5, one infinite family of 2-tight nus integers with D_1(N) – D(N) ≥ 1 and one infinite family of 3-tight nus integers with D_1 (N) – D(N) > 1.