Exploring Oriented Threshold Graphs: A Study on Controllability/Observability

In this letter, we explore the controllability/observability of Laplacian networks on oriented threshold graphs (OTGs). We present the spectrum and modal matrix associated with their out-degree Laplacian matrices. Our analysis shows that the out-degree Laplacian matrix is diagonalizable, allowing us to establish necessary and sufficient conditions for Laplacian controllability. Furthermore, we demonstrate that with a binary input matrix, the minimum number of control signals required for controllability equals the maximum geometric multiplicity of out-degree Laplacian eigenvalues. The results also hold for the observability of OTGs with in-degree Laplacian matrices.