Laplacian eigenvalue allocation by asymmetric weight assignment for path graphs

In this paper we afford the Laplacian eigenvalue allocation problem for path graphs through the choice of asymmetric weights. After a brief introduction on the meaningfulness of this mathematical setting in several control problems, and more specifically in distributed control of multi-agent systems and robotic networks, the solution is sought both analytically and through an algorithm. Moreover, it is proved that, if the eigenvalues are chosen satisfying the interlacing property, then the solution is positive. Some examples showing the effectiveness of the proposed solution close the paper.