NDLI: Minimal Laplacian controllability problems of threshold graphs

Content Provider	IET Digital Library
Author	Hsu, Shun Pin
Abstract	This study is concerned with the minimal controllability problem of a connected threshold graph following the Laplacian dynamics. The goal is to find the minimum number of controllers and a small set of vertices for the controllers to connect to render the graph Laplacian controllable. A simple algorithm is provided 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 make a connected threshold graph Laplacian controllable is the maximum multiplicity of Laplacian eigenvalues of the graph, and this minimum can be achieved using a binary control matrix. If a controller can be connected to one vertex only, the minimum number is the difference between the number of vertices in the graph and the number of vertices with different degrees. The condition also implies that the controllers ensuring the Laplacian controllability should be connected to the vertices with repeating degrees to break the symmetry of the network topology. Several examples are provided to illustrate the authors' results.
Starting Page	1639
Ending Page	1645
Page Count	7
ISSN	17518644
Volume Number	13
e-ISSN	17518652
Issue Number	Issue 11, Jul (2019)
Alternate Webpage(s)	https://digital-library.theiet.org/content/journals/iet-cta/13/11
Alternate Webpage(s)	https://digital-library.theiet.org/content/journals/10.1049/iet-cta.2018.5875
Journal	IET Control Theory & Applications
Publisher Date	2019-04-16
Access Restriction	Open
Rights Holder	© The Institution of Engineering and Technology
Subject Keyword	Algebra Binary Control Matrix Combinatorial Mathematics Connected Threshold Graph Laplacian Controllable Controllability Differential Equation (numerical Analysis) Eigen Values And Eigen Function Graph Theory Laplace Equation Laplacian Dynamics Laplacian EigenValue Laplacian Matrix Matrix Algebra Maximum Multiplicity Minimal Laplacian Controllability Problem Minimum Number Necessary Condition Network Topology Orthogonal Laplacian EigenVector Sufficient Condition
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Control and Systems Engineering Human-Computer Interaction Electrical and Electronic Engineering Computer Science Applications

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