Minimal Laplacian controllability problems of threshold graphs

Abstract

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.

A Sylvester-Kac matrix type and the Laplacian controllability of half graphs

“…In [29] one can find how the Laplacian spectrum and eigenspaces of a threshold graph are modified by (0, 1)-operations in its binary generating procedure. Also [21] explains how to identify the Laplacian eigenvalues of a threshold graph from its Laplacian matrix. Finally, we are in the position to prove the main result of this section.…”