  This paper is concerned with the controllability problem of a connected
threshold graph following the Laplacian dynamics. An algorithm is proposed to
generate a spanning set of orthogonal Laplacian eigenvectors of the graph from
a straightforward computation on its Laplacian matrix. A necessary and
sufficient condition for the graph to be Laplacian controllable is then
proposed. The condition suggests that the minimum number of controllers to
render a connected threshold graph controllable is the maximum multiplicity of
entries in the conjugate of the degree sequence determining the graph, and this
minimum can be achieved by a binary control matrix. The second part of the work
is the introduction of a novel class of single-input controllable graphs, which
is constructed by connecting two antiregular graphs with almost the same size.
This new connecting structure reduces the sum of the maximum vertex degree and
the diameter by almost one half, compared to other well-known single-input
controllable graphs such as the path and the antiregular graph, and has
potential applications in the design of controllable graphs subject to
practical edge constraints. Examples are provided to illustrate our results.
