Controllability of undirected graphs

In control theory, networked dynamical systems have a wide range of engineering applications. In a relational graph among followers (F)(F) and leaders (R)(R), new necessary and sufficient conditions for a pair (F,R)(F,R) to be controllable are presented. The choice of leader vertices for controllability is shown to be facilitated by identifying the core vertices associated with the eigenvectors of a matrix S related to a graph. We present new necessary and sufficient conditions for a graph to be controllable relative to its adjacency matrix or to its signless Laplacian without having to evaluate any eigenspaces, which is the criterion usually employed. The symmetries of the system graph represented by S are also shown to aid in the choice of a potential leader vertex that is able to control the follower subgraph on its own. Moreover, we define k-omnicontrollable graphs for controllability by any k leaders and show that simple 1-omnicontrollable graphs have only two possible automorphism groups.