Laplacian Controllability of Symmetric Directed Path and Ring Graphs

This paper investigates the Laplacian controllability of symmetric directed path and ring graphs, which are prevalent in real-world network applications. The analysis focuses on deriving explicit expressions for the eigenvalues and eigenvectors of the Laplacian matrices associated with these graphs. Based on these expressions, necessary and sufficient conditions for Laplacian controllability are established. The theoretical results are illustrated with practical examples, demonstrating the application of the controllability conditions. The findings of this study contribute to a deeper understanding of network controllability and provide valuable insights for designing and manipulating real-world networks with symmetric directed path and ring topologies.