STP Approach to Formulate k-Externally Stable Sets of Graphs

In this paper the structure of graphs in terms of k-externally stable set (k-ESS) is investigated based on a new matrix product, called semi-tensor product of matrices. By defining an eigenvector and an eigenvalue of node subset of a graph, three necessary and sufficient conditions of k-ESS are proposed in a matrix form. Using these conditions, the concepts of k-ESS matrix are introduced. The k-ESS matrix provides the complete information of the corresponding structures of a graph. Further, an algorithm is designed to find all the structures of a graph by conducting a series of matrix operation. The proposed method and results may offer a new way to investigate the problems related to graph structures in the field of network systems.