连通图的Laplacian能控性
Laplacian Controllability of Interconnected Graphs
摘要：
在这项工作中，我们考虑的Laplacian可控图的有限个单输入Laplacian可控图的互连构造的图。我们首先研究了由两个连通的单图（称为结构图和胞图）的合成图实现的互连。假设细胞图是拉普拉斯控制的输入连接到一些特殊的顶点称为复合顶点。复合图是由构成结构图的复合顶点将所有的细胞图连接起来而构成的。然后，我们证明了结构图是拉普拉斯控制的输入连接到一些顶点的图的当且仅当复合图是拉普拉斯控制的输入连接到该复合顶点。在本文的第二部分中，我们把路看作是由有限个两点反正则图互连而成的图，并且可能连接到一条单点路，其中反正则图的两个顶点被解释为终端点（或控制点）和度重复点。我们证明了在一个类似的连接方案下，如果我们增加反正则图的顶点数和路的顶点数，单输入Laplacian能控性是保持的，并给出了数值例子来说明我们的结果.
英文摘要：
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.
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