Double commutative-step digraphs with minimum diameters

Esqué, P.; Aguiló, Francesc; Fiol, M. A.

doi:10.1016/0012-365x(93)90363-x

Cited by 39 publications

(37 citation statements)

References 12 publications

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“…From now on, we simplify "weighted 2-Cayley digraph G(c; a, b; W a , W b )" to "digraph G(c; a, b; W a , W b )". Metrical properties of these digraphs have been widely studied using "minimum distance diagrams" [7,6,11]. j]] is related to the equivalence class [ia + jb] c , and we denote the weight of the unit square [[i, j]] by δ(i, j) = iW a + jW b .…”

Section: L-shaped Minimum Distance Diagramsmentioning

confidence: 99%

“…Their generalization to weighted 2-Cayley digraphs, adding weights to the arcs, allows other applications to be studied. Looking for paths of minimum length in these digraphs, periodical plane tessellations with L-shaped tiles appear in the bibliography [7,6]. Particular classes of L-shapes, so called minimal distance diagrams, have been used to study certain distance properties in 2-Cayley digraphs, and have also been associated to numerical 3-semigroups [10,2].…”

Section: Introductionmentioning

confidence: 99%

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Classification of numerical 3-semigroups by means of L-shapes

Aguiló

Marijuán

2014

Semigroup Forum

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We recall L-shapes, which are minimal distance diagrams, related to weighted 2-Cayley digraphs, and we give the number and the relation between minimal distance diagrams related to the same digraph. On the other hand, we consider some classes of numerical semigroups useful in the study of curve singularity. Then, we associate L-shapes to each numerical 3-semigroup and we describe some main invariants of numerical 3-semigroups in terms of the associated L-shapes. Finally, we give a characterization of the parameters of the L-shapes associated to a 3-numerical semigroup in terms of its generators, and we use it to classify the numerical 3-semigroups of interest in curve singularity.

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Section: L-shaped Minimum Distance Diagramsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Classification of numerical 3-semigroups by means of L-shapes

Aguiló

Marijuán

2014

Semigroup Forum

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“…Li, et al [1] and Esqué, et al [12] gave many infinite families of 0-TODLN and 1-TODLN. A problem was proposed in [1]: for any given k 0, is there an infinite family of k-TODLN?…”

Section: Introductionmentioning

confidence: 98%

The construction of infinite families of any k-tight optimal and singular k-tight optimal directed double loop networks

Chen

Meng

et al. 2007

SCI CHINA SER A

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The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is: for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained: (1) for any k 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k, e, c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k 0, an infinite family of singular k-tight optimal DLN can be constructed.

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“…For more details, see Esqué, Aguiló and Fiol [5], Aguiló and Fiol [1], and Fiol, Yebra, Alegre and Valero [6].…”

Section: Double-step Digraphsmentioning

confidence: 99%

“…In the case of the double-step digraphs also with the standard diameter, Morillo, Fiol and Fàbrega [10] solved the (∆, D) problem and provided some infinite families of digraphs which solve the (∆, N ) problem for their corresponding numbers of vertices. Esqué, Aguiló and Fiol [5], and Aguiló and Fiol [1] contributed to the second problem with more general results. Double-step digraphs were proposed and studied as models for the so-called 'local area networks', in which several computers placed at short distances exchange data at very high speed, as explained in Fiol, Yebra, Alegre and Valero [6].…”

Section: Introductionmentioning

confidence: 99%

The (Delta,D) and (Delta,N) problems in double-step digraphs with unilateral distance

Dalfó¹,

Fiol²

2014

ejgta

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We study the (∆, D) and (∆, N ) problems for double-step digraphs considering the unilateral distance, which is the minimum between the distance in the digraph and the distance in its converse digraph, the latter obtained by changing the directions of all the arcs.The first problem consists of maximizing the number of vertices N of a digraph, given the maximum degree ∆ and the unilateral diameter D * , whereas the second one (somehow dual of the first) consists of minimizing the unilateral diameter given the maximum degree and the number of vertices. We solve the first problem for every value of the unilateral diameter and the second one for infinitely many values of the number of vertices. Moreover, we compute the mean unilateral distance of the digraphs in the families considered.

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Double commutative-step digraphs with minimum diameters

Cited by 39 publications

References 12 publications

Classification of numerical 3-semigroups by means of L-shapes

Classification of numerical 3-semigroups by means of L-shapes

The construction of infinite families of any k-tight optimal and singular k-tight optimal directed double loop networks

The (Delta,D) and (Delta,N) problems in double-step digraphs with unilateral distance

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