Laplacian controllable graphs based on connecting two antiregular graphs

Hsu, Shun‐Pin; Yang, Ping-Yen

doi:10.1049/iet-cta.2018.5484

Cited by 8 publications

(5 citation statements)

References 26 publications

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“…Now we present a systematic approach to generate a set of

k - 1

eigenvectors of

{scriptL}_{T}^{false(k false)}

. It has been applied to calculate the Laplacian eigenvectors of a special class of threshold graphs [26–28]. We will show in the following that it is actually applicable to the entire class of threshold graphs.…”

Section: Resultsmentioning

confidence: 99%

“…This result was later extended to the multi‐input case where more vertices with the same degree were considered [27]. The Laplacian controllability of hybrid graphs constructed by connecting one antiregular graph to another antiregular graph [28] or to a path graph [29] was studied. In this work, we consider the case of generally connected threshold graphs and address their minimal Laplacian controllability issues.…”

Section: Introductionmentioning

confidence: 99%

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Minimal Laplacian controllability problems of threshold graphs

Hsu

2019

IET Control Theory & Applications

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This paper is concerned with the controllability problem of a connected threshold graph following the Laplacian dynamics. An algorithm is proposed to generate a spanning set of orthogonal Laplacian eigenvectors of the graph from a straightforward computation on its Laplacian matrix. A necessary and sufficient condition for the graph to be Laplacian controllable is then proposed. The condition suggests that the minimum number of controllers to render a connected threshold graph controllable is the maximum multiplicity of entries in the conjugate of the degree sequence determining the graph, and this minimum can be achieved by a binary control matrix. The second part of the work is the introduction of a novel class of single-input controllable graphs, which is constructed by connecting two antiregular graphs with almost the same size. This new connecting structure reduces the sum of the maximum vertex degree and the diameter by almost one half, compared to other well-known single-input controllable graphs such as the path and the antiregular graph, and has potential applications in the design of controllable graphs subject to practical edge constraints. Examples are provided to illustrate our results.

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“…Now we present a systematic approach to generate a set of

k - 1

eigenvectors of

{scriptL}_{T}^{false(k false)}

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimal Laplacian controllability problems of threshold graphs

Hsu

2019

IET Control Theory & Applications

Self Cite

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“…A natural issue raised by our results is that whether the property of single-input Laplacian controllability can be preserved by interconnecting antiregular graphs with different numbers of vertices? It was shown that the property is not preserved in connecting a four-vertex antiregular graph with a two-vertex path through their respective terminal vertices [19], but it is preserved if the two-vertex path is connected to the degree-repeating vertex of the four-vertex antiregular graph [20]. How to interconnect a finite number of different antiregular graphs to preserve the property is definitely worthy of a separate note.…”

Section: Discussionmentioning

confidence: 99%

“…An important insight into the discovery is that interconnecting two single‐input Laplacian controllable graphs appropriately might preserve the single‐input controllability. The first member features the interconnection of a path and an antiregular graph [19], and the second, the interconnection of two antiregular graphs [20]. An antiregular graph is a connected simple graph that has exactly one pair of degree‐repeating vertices, or the vertices that have the same number of neighbouring vertices [21].…”

Section: Introductionmentioning

confidence: 99%

Generalising Laplacian controllability of paths

Hsu

Yang

2019

IET Control Theory & Applications

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It is well known that if a network topology is a path or line and the states of vertices or nodes evolve according to the consensus policy, then the network is Laplacian controllable by an input connected to its terminal vertex. In this work a path is regarded as the resulting graph after interconnecting a finite number of two-vertex antiregular graphs and then possibly connecting one more vertex. It is shown that the single-input Laplacian controllability of a path can be extended to the case of interconnecting a finite number of k-vertex antiregular graphs with or without one more vertex appended, for any positive integer k. The methods to interconnect these antiregular graphs and to select the vertex for connecting the single input that renders the network Laplacian controllable are presented as well.

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“…For n ≥ 2, denote by A n and A n the connected antiregular and disconnected antiregular, respectively, graphs of order n. It needs to be mentioned here that connected antiregular graphs were referred as pairlone graphs in [57] and half-complete graphs in [29]. Antiregular graphs have several interesting properties, and the connected antiregular graphs have found some applications in control theory [38][39][40]. Also, antiregular graphs are sometimes considered as the graphs opposite to the regular graphs [1,13,15,34].…”

Section: Introductionmentioning

confidence: 99%

A survey of antiregular graphs

2020

Contrib. Math.

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The set of all different degrees of the vertices of a graph G is known as the degree set of G. A nontrivial graph of order n whose degree set consists of n−1 elements is called an antiregular graph. Antiregular graphs have been studied in literature also under other names, including "quasi-perfect graphs", "maximally nonregular graphs" and "degree antiregular graphs". This paper aims to gather the known results concerning the antiregular graphs.

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Laplacian controllable graphs based on connecting two antiregular graphs

Cited by 8 publications

References 26 publications

Minimal Laplacian controllability problems of threshold graphs

Minimal Laplacian controllability problems of threshold graphs

Generalising Laplacian controllability of paths

A survey of antiregular graphs

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