A symbolical algorithm on additive basis and double-loop networks

A double-loop digraph G = G(N; s1, s2), with gcd(N, s1, s2) = 1, has the set of vertices V = ZN and the adjacencies are given by u → u + si (mod N) i = 1, 2. The diameter of G, denoted by D(N; s1, s2), is known to be lower bounded by lb(N) with D(N) = min1 ≤ s1 < s2 < N, gcd(N, s1, s2)=1 D(N; s1, s2) ≥ ⌈ √3N ⌉ - 2 = lb(N). This lower bound is known to be sharp. For a fixed N ∈ N, some algorithms to find D(N) and steps 1 ≤ α < β < N such that D(N; α, β) = D(N) are known through the bibliography, being of numerical nature all of them.In this paper we propose a symbolic algorithm on the following problem: Given a number of vertices N0 ∈ N, find if possible an infinite family of tiqht double-loop digraphs G(x) = G(N(x); s1(x), s2(x)) such that N(x0) = N0 and D(x) = D(N(x); s1(x),s2(x)) = lb(N(x)) ∀x ≥ x0. This family is parameterized by the integer x with N(x) ∈ Z and s1(x); s2(x) ∈ Z/(N(x)). As a direct consequence of such an explicit family of digraphs G(x), we also have an additive basis {s1(x), s2(x)} which covers the elements of Z/(N(x)) with optimal order.The time cost of this algorithm is O(√N0 log N0), in the worst case.