Optimal double-loop networks with non-unit steps∗

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<s2<N, gcd(N,s1,s2)=1 D(N ; s1, s2), D1(N) = min 1<s<N D(N ; 1, s). Some early works about the diameter of these digraphs studied the minimization of D(N ; 1, s), for a fixed value N , with 1 < 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.