New dense families of triple loop networks

Abstract

Multi-loop digraphs are widely studied mainly because of their symmetric properties and their applications to loop networks. A multi-loop digraph G = G( N; s 1, …, s A ) has set of vertices V = Z N and adjacencies given by v → v + s i mod N, i = 1, …, Δ. For every fixed N, an usual extremal problem is to find the minimum value = min s 1…s δ∈ Z N D(N;s 1…sδ where D( N; s 1, …, s A ) is the diameter of G. A closely related problem is to find the maximum number of vertices for a fixed value of the diameter. For Δ = 2, all optimal families have been found by using a geometrical approach. For Δ = 3, only some dense (but possibly not optimal) families are known. In this work some new dense families are given for Δ = 3 using a geometrical approach. This technique was already adopted in several papers for Δ = 2 (see for instance [7, 10]). These families improve all previous known results.