Controllability of the multi-agent system modeled by the threshold graph with one repeated degree

The controllability issue of multi-agent systems modeled by a special class of graphs is studied. Suppose the modeling graph is simple, connected and driven by only one controller. The author shows that the system is controllable if the graph has exactly two vertices with the same degree and either one of these two vertices is selected to receive the control input. This result is extended to the case of the threshold graph that has only one repeated degree. Specifically, it is shown that if the threshold graph has only one repeated degree and its multiplicity is m, then m−1 is the minimum number of the controllers required to make the system controllable. In addition, the necessary and sufficient condition that characterizes the binary control vectors to ensure the controllability of the system is derived.