Iterated line digraphs are asymptotically dense

Dalfó, C.

doi:10.1016/j.laa.2017.04.036

Cited by 5 publications

(9 citation statements)

References 3 publications

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“…Observe that if this probability is not zero, then the destination vertex z must belong to S ⋆ 4 (v) and also to S ⋆ 6 (w). Therefore, since S ⋆ 4 (v) ⊆ S 4 (v) and (6). We can verify from the sequence representations of v and w that the following chain of inclusions hold:…”

Section: And Only If There Exists a Permutation σ Of The Symbol Alphabetmentioning

confidence: 84%

Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing (with examples and remarks)

Fàbrega¹,

Martí-Farré²,

Muñoz³

2022

Preprint

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In this paper, we present a detailed study of the reach distance-layer structure of the De Bruijn and Kautz digraphs, and we apply our analysis to the performance evaluation of deflection routing in De Bruijn and Kautz networks. Concerning the distance-layer structure, we provide explicit polynomial expressions, in terms of the degree of the digraph, for the cardinalities of some relevant sets of this structure. Regarding the application to defection routing, and as a consequence of our polynomial description of the distance-layer structure, we formulate explicit rational expressions, in terms of the degree of the digraph, for some probabilities of interest in the analysis of this type of routing.De Bruijn and Kautz digraphs are fundamental examples of digraphs on alphabet and iterated line digraphs. If the topology of the network under consideration corresponds to a digraph of this type, we can perform, in principle, a similar vertex layer description.

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Section: And Only If There Exists a Permutation σ Of The Symbol Alphabetmentioning

confidence: 84%

Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing (with examples and remarks)

Fàbrega¹,

Martí-Farré²,

Muñoz³

2022

Preprint

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“…If G is a strongly connected d-regular digraph, different from a directed cycle, with diameter D, then its line digraph L k G is d-regular with n k = d k n vertices and has (asymptotically optimal) diameter D + k. In fact, for a strongly connected general digraph, the first author [5] proved that the iterated line digraphs are always asymptotically dense. For more details, see Harary and Norman [8], Aigner [1], and Fiol, Yebra, and Alegre [7].…”

Section: Preliminariesmentioning

confidence: 99%

“…For instance, the 2-Fibonacci digraphs F (2, k) with k ≤ 4 and 2, 3, 5, 8 vertices, are shown in Figure 1, whereas the 1-Fibonacci digraphs F (d, 1) on d vertices, with d ∈ [2,5], are depicted in Figure 3.…”

Section: D-fibonacci Digraphs On Alphabetsmentioning

confidence: 99%

On d-Fibonacci digraphs

Dalfó

Fiol

2021

EJGTA

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The d-Fibonacci digraphs F (d, k), introduced here, have the number of vertices following some generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, F (d, k) has diameter d + k − 2 and is semi-pancyclic; that is, it has a cycle of every length between 1 and , with ∈ {2k − 2, 2k − 1}. Moreover, it turns out that several other numbers of F (d, k) (of closed l-walks, classes of vertices, etc.) also follow the same linear recurrences as the numbers of vertices of the d-Fibonacci digraphs.

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“…In [6] it was proven that a digraph G is L-divergent if and only if at least one strong component of G is not a cycle or G has at least two cycles joined by a path. The class of L-divergent digraphs includes all strongly connected digraphs other than a cycle, which have been proven to be asymptotically dense in [17] for the regular case and in [12] for irregular digraphs.…”

Section: If G Does Not Have Any Loops Then For Everymentioning

confidence: 99%

Zero forcing in iterated line digraphs

Ferrero

Kalinowski

Stephen

2019

Discrete Applied Mathematics

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Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks.In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications.1 in network science, it models the spread of a disease over a population, or of an opinion in a social network (see [14]).In addition to its intrinsic relation to minimum rank, zero forcing is closely related to power domination, a graph theory concept introduced in [20] to optimize the monitoring process of electrical power networks. From the definitions of power domination and zero forcing, it follows that the closed out neighborhood of a power dominating set is a zero forcing set, and a stronger relationship between zero forcing and power domination was established in [7].Zero forcing, minimum rank and power domination are all NP-hard problems, as proven in [1], [10], and [20], respectively. Thus, it is important to obtain bounds for the minimum rank, the zero forcing and the power domination numbers, as well as closed formulas to calculate them for families of graphs. Although power domination and zero forcing where introduced on undirected graphs, they were extended to digraphs in [1] and [5], respectively. Further results on zero forcing on digraphs can be found in [5], [15], [21], [23] and [31]. Power domination in digraphs has not been so thoroughly explored, but recently zero forcing and power domination for de Bruijn and Kautz digraphs was studied in [19]. De Bruijn and Kautz digraphs are iterated line digraphs of the complete digraph, with and without loops, respectively. In this work, we extend the results in [19] to zero forcing and power domination of iterated line digraphs of any regular digraph.The line digraph has been used in a broad range of disciplines, but its large number of applications precludes us from including an exhaustive summary here. In the context of this work, since the line digraph of a digraph described by a unitary matrix can also be described by a unitary matrix, iterated line digraphs are used to obtain arbitrarily large digraphs described by unitary matrices (see [24] and [28]). Such digraphs are frequently used to model quantum systems in physics, chemistry, and engineering (see [22]), and...

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Iterated line digraphs are asymptotically dense

Cited by 5 publications

References 3 publications

Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing (with examples and remarks)

Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing (with examples and remarks)

On d-Fibonacci digraphs

Zero forcing in iterated line digraphs

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