Strict Lyapunov functions for consensus under directed connected graphs

It is known that for consensus of systems interconnected under a general directed graph topology a necessary and sufficient condition for consensus is that there exist at least one rooted spanning tree. In this paper we present an original statement of linear algebra that serves to characterise the spanning-tree condition for directed graphs in terms of a Lyapunov equation involving the graph's Laplacian. Our results apply to the case of systems described by simple first and second order integrators. As a result, we provide strict Lyapunov functions that ensure, via direct constructive proof, global exponential stability of the consensus manifold.