Triple-loop networks with arbitrarily many minimum distance diagrams

Abstract

Minimum distance diagrams are a way to encode the diameter and routing information of multi-loop networks. For the widely studied case of double-loop networks, it is known that each network has at most two such diagrams and that they have a very definite form "L-shape''. In contrast, in this paper we show that there are triple-loop networks with an arbitrarily big number of associated minimum distance diagrams. For doing this, we build-up on the relations between minimum distance diagrams and monomial ideals.Comment: 17 pages, 8 figure

An algorithm to compute the primitive elements of an embedding dimension three numerical semigroup

“…When n = 3 they are well known as L-shaped tiles and many works appeared in the bibliography ([1,6]) on Cayley digraphs and numerical semigroups. Less is known of generic tiles when n ≥ 4, with the exception of some particular cases ( [7,10]). …”

“…Let us define the discrete cone of unitary cubes Minimum distance diagrams were used first in the computation of distances in weighted and non-weighted Cayley digraphs ( [10]). These diagrams tesellate the R n−1 space (so they are also named tiles).…”

Connectedness of finite distance graphs

“…They identified paths starting from node 0 with pairs of integers and associated with every path its destination node. These diagrams and their extensions to higher dimensions (graphs with more than two jumps) have aroused interest both from the pure combinatorial point of view and because of its applications in optimal network design [7,12,6,9]. Let us reproduce an analog construction in the case of a two‐jump (and undirected) circulant graph C n ( s 1 , s 2 ).…”

On the number of L-shapes in embedding dimension four numerical semigroups

“…A unified definition of minimum distance diagrams was given by P. Sabariego and F. Santos in 2009 [9, Definition 2.1] although other authors used this concept, see for instance Fiol et al [5] and Rödseth [7]. Following the definition of [9], a minimum distance diagram H related to G is a connected set of N unit cubes in R k with different vertex label and the following two properties (1) if u = [[i 1 , . .…”

“…When k = 2, it has been shown that these digraphs have two related MDD at most. For k = 3, Sabariego and Santos [9] gave an infinite family of digraphs with many associated MDDs. More precisely, given t ≡ 0 (mod 3), set m = 2 + t + t 2 ; then, the digraph G t = C(m(m − 1); 1 + m, 1 + mt, 1 + mt 2 ; 1, 1, 1) has 3(t + 2) associated MDDs.…”