  In this work we consider the Laplacian controllability of a graph constructed
by interconnecting a finite number of single-input Laplacian controllable
graphs. We first study the interconnection realized by the composite graph of
two connected simple graphs called the structure graph and the cell graph.
Suppose the cell graph is Laplacian controllable by an input connected to some
special vertex called the composite vertex. The composite graph is constructed
by interconnecting all cell graphs through the composite vertices which alone
form the structure graph. We then show that the structure graph is Laplacian
controllable by an input connected to some vertex of the graph if and only if
the composite graph is Laplacian controllable by that input connected to that
composite vertex. In the second part of the paper, we view a path as a graph
generated by interconnecting a finite number of two-vertex antiregular graph,
and possibly connected to a one-vertex path, where the two vertices of the
antiregular graph are interpreted as the terminal vertex (or the dominating
vertex) and the degree-repeating vertex. We show that with a similar connecting
scheme the single-input Laplacian controllability is preserved if we increase
the number of vertices of the antiregular graph and that of the path. Numerical
examples are presented to illustrate our results.
