Laplacian Controllability of Threshold Graphs

TL;DR

This paper investigates the controllability of threshold graphs under Laplacian dynamics, providing an algorithm for eigenvector generation, a necessary and sufficient controllability condition, and introducing a new class of single-input controllable graphs with improved structural properties.

Contribution

It offers a novel controllability condition for threshold graphs, an eigenvector generation algorithm, and introduces a new class of single-input controllable graphs with optimized structural features.

Findings

Minimum controllers equal the maximum multiplicity of degree sequence entries.

A binary control matrix suffices to achieve controllability.

New graph structures reduce degree and diameter, enhancing controllability.

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.