Laplacian Controllability Classes for Threshold Graphs

Let G be a graph on n vertices with Laplacian matrix L and let b be a binary vector of length n. The pair (L,b) is controllable if the smallest L-invariant subspace containing b is of dimension n. The graph G is called essentially controllable if (L,b) is controllable for every b / ∈ ker(L), completely uncontrollable if (L,b) is uncontrollable for every b, and conditionally controllable if it is neither essentially controllable nor completely uncontrollable. In this paper, we completely characterize the graph controllability classes for threshold graphs. We first observe that the class of threshold graphs contains no essentially controllable graph. We prove that a threshold graph is completely uncontrollable if and only if its Laplacian matrix has a repeated eigenvalue. In the process, we fully characterize the set of conditionally controllable threshold graphs.