LAUSR

In this letter, the directed threshold graphs (DTGs) and their Laplacian controllability issues are studied. The graphs are constructed by a sequence of graph operations that include ‘union’ and ‘directed join’, where the directed join means in adding a new node, the direction of the new edges are either from the new node to all existing ones or the other way around. It is shown that the Laplacian spectra of the graphs can be readily identified and Laplacian eigenspaces fully characterized, and thus the binary control matrices can be determined to render the graphs Laplacian controllable. In particular, simple algorithms are proposed to facilitate the Laplacian eigenspace analysis and to select the nodes to which the minimum number of controllers can be connected to ensure the Laplacian controllability. Examples are provided to illustrate the proposed results.