Algebraic Expression and Construction of Control Sets of Graphs Using Semi-Tensor Product of Matrices

Using a new matrix analysis tool, called semi-tensor product of matrices (STP) developed in recent years, this paper investigates the problem of finding control sets (or dominating sets) of graphs mathematically. By defining characteristic vectors for vertex subsets of graphs, three sufficient and necessary conditions of control sets are proposed, based on which an algebraic algorithm that can find all the control sets of a graph is established. Further, the concepts of k-capacity and k-harmony control sets of graphs are proposed, which can model some real-world problems such as security monitoring of streets and wireless sensor networks. Several sufficient and necessary conditions are established to judge whether or not a vertex subset is a k-capacity control set; a necessity that a vertex subset is a k-harmony control set is proposed, which facilitates finding all the k-harmony control sets of graphs. The correctness and effectiveness of the results is finally examined in detail by examples. The approach of this paper may provide a new angle and means to understand and analyze the structure of graphs.