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Laplacian Controllability of Interconnected Graphs

We consider the controllability of a graph constructed by interconnecting a finite number of single-input Laplacian controllable (SILC) graphs. We first study the interconnection realized by the composite graph of two SILC graphs, called the structure graph and the cell graph, respectively. 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 several identical cell graphs through the composite vertices such that connecting these composite vertices alone forms the structure graph. We prove that the structure graph is SILC by an input connected to some vertex of the graph if and only if the composite graph is SILC by the same input connected to the same vertex. The second part of this article generalizes the path structure by viewing it as a serial interconnection of two-vertex antiregular graphs, with or without appending a terminal one-vertex path. We show that its SILC property is preserved if we increase the number of vertices of the antiregular graphs and/or that of the terminal path. Examples are provided to illustrate the novel class of SILC graphs we propose.