  In this technical note, we study the controllability of diffusively coupled
networks from a graph theoretic perspective. We consider leader-follower
networks, where the external control inputs are injected to only some of the
agents, namely the leaders. Our main result relates the controllability of such
systems to the graph distances between the agents. More specifically, we
present a graph topological lower bound on the rank of the controllability
matrix. This lower bound is tight, and it is applicable to systems with
arbitrary network topologies, coupling weights, and number of leaders. An
algorithm for computing the lower bound is also provided. Furthermore, as a
prominent application, we present how the proposed bound can be utilized to
select a minimal set of leaders for achieving controllability, even when the
coupling weights are unknown.
