Laplacian Controllability of Interconnected Graphs

TL;DR

This paper investigates the conditions under which interconnected graphs, constructed from controllable subgraphs, maintain Laplacian controllability, with specific focus on composite graphs and antiregular graph structures.

Contribution

It establishes a necessary and sufficient condition for Laplacian controllability of composite graphs built from controllable subgraphs and analyzes controllability preservation in interconnected antiregular graphs.

Findings

Laplacian controllability of composite graphs depends on the structure graph's controllability.

Controllability is preserved when increasing the size of antiregular graphs and paths.

Numerical examples validate the theoretical results.

Abstract

In this work we consider the Laplacian controllability of a graph constructed by interconnecting a finite number of single-input Laplacian controllable graphs. We first study the interconnection realized by the composite graph of two connected simple graphs called the structure graph and the cell graph. Suppose the cell graph is Laplacian controllable by an input connected to some special vertex called the composite vertex. The composite graph is constructed by interconnecting all cell graphs through the composite vertices which alone form the structure graph. We then show that the structure graph is Laplacian controllable by an input connected to some vertex of the graph if and only if the composite graph is Laplacian controllable by that input connected to that composite vertex. In the second part of the paper, we view a path as a graph generated by interconnecting a finite number of two-vertex antiregular graph, and possibly connected to a one-vertex path, where the two vertices of the antiregular graph are interpreted as the terminal vertex (or the dominating vertex) and the degree-repeating vertex. We show that with a similar connecting scheme the single-input Laplacian controllability is preserved if we increase the number of vertices of the antiregular graph and that of the path. Numerical examples are presented to illustrate our results.