On Dense 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, \(G = G(N; s_1, \ldots, s_\Delta)\) with \(1 \leq s_1 < \cdots < s_\Delta \leq N-1\) and \(\gcd(N, s_1, \ldots, s_\Delta) = 1\), has set of vertices \(V ={Z}_N\) and adjacencies given by \(v \mapsto v + s_i \mod N, i = 1, \ldots, \Delta\). For every fixed \(N\), an usual extremal problem is to find the minimum value \[D_\Delta(N)=\min\limits_{s_1,\ldots,s_\Delta \in Z_N}(N; s_1, \ldots, s_\Delta)\] where \(D(N; s_1, \ldots, s_\Delta)\) 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 \(\Delta = 2\), all optimal families have been found by using a geometrical approach. For \(\Delta = 3\), only some dense families are known. In this work, a new dense family is given for \(\Delta = 3\) using a geometrical approach. This technique was already adopted in several papers for \(\Delta = 2\) (see for instance [5, 7]). This family improves the dense families recently found by several authors.