Controllability of undirected graphs

Farrugia, Alexander; Sciriha, Irene

doi:10.1016/j.laa.2014.04.022

Cited by 16 publications

(14 citation statements)

References 6 publications

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“…Before we introduce U-controllable graphs, we summarize various results from control theory, on which the definition of U-controllable graphs is based. These results are also mentioned in [6]. A networked dynamical system is a set of n agents and a set of links interconnecting these agents.…”

Section: Theorem 24 the Number Of Orbits Of Any Automorphism Of A Gmentioning

confidence: 73%

“…We have the following two corollaries to Theorem 4.4, which were originally proved in [6] (Theorems 4.3 and 4.4) in a different manner. As mentioned earlier, a relation between the number of main eigenvalues of the matrices A and Q associated with the same graph G is not known.…”

Section: A-controllable and Q-controllable Graphsmentioning

confidence: 99%

“…In the rest of this section, we shall determine necessary and sufficient conditions for a graph to be U-controllable. To this end, we first combine several results that were presented in [6] in the theorem below. We require (U, j) to be a controllable pair.…”

Section: U-controllable Graphsmentioning

confidence: 99%

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On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs

Farrugia

Sciriha

2015

ELA

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Abstract. A universal adjacency matrix U of a graph G is a linear combination of the 0-1 adjacency matrix A, the diagonal matrix of vertex degrees D, the identity matrix I and the matrix J each of whose entries is 1. A main eigenvalue of U is an eigenvalue having an eigenvector that is not orthogonal to the all-ones vector. It is shown that the number of distinct main eigenvalues of U associated with a simple graph G is at most the number of orbits of any automorphism of G. The definition of a U-controllable graph is given using control-theoretic techniques and several necessary and sufficient conditions for a graph to be U-controllable are determined. It is then demonstrated that U-controllable graphs are asymmetric and that the converse is false, showing that there exist both regular and non-regular asymmetric graphs that are not U-controllable for any universal adjacency matrix U. To aid in the discovery of these counterexamples, a γ-Laplacian matrix L (γ) is used, which is a simplified form of U. It is proved that any U-controllable graph is a L (γ)-controllable graph for some parameter γ.

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Section: Theorem 24 the Number Of Orbits Of Any Automorphism Of A Gmentioning

confidence: 73%

Section: A-controllable and Q-controllable Graphsmentioning

confidence: 99%

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On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs

Farrugia

Sciriha

2015

ELA

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“…. , W n (G) are all non-singular is termed omnicontrollable in [5] and has the property that none of its eigenvectors have zero entries. Hence, the probability that a graph is omnicontrollable tends to 1 as its number of vertices increases.…”

Section: Final Remarksmentioning

confidence: 99%

“…Several papers on the subject of controllable graphs have appeared in the literature, most notably [3][4][5]7,16,18]. For the link between controllable graphs and control theory, the reader is referred to [3][4][5]7,18]. We shall summarize the main results on controllable graphs known in the literature that are relevant to this paper in the next section.…”

mentioning

confidence: 99%