Optimal double-loop networks with non-unit steps

Abstract

A double-loop digraph G(N; s1, s2) = G(V,E) is defined by V = ℤN and E = {(i, i + s1), (i, i + s2)| i ∈ V}, 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 1≤s1 < s < N. Although the identity D(N) = D1(N) holds for infinite values of N, there are also another infinite set of integers with D(N) < D1(N). These other integral values of N are called non-unit step integers or nus integers. In this work we give a characterization of nus integers and a method for finding infinite families of nus integers is developed. Also the tight nus integers are classified. As a consequence of these results, some errata and some flaws in the bibliography are corrected.