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

中文翻译：

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使用 STP 理论制定和搜索图的 k-ESS 的矩阵方法

在本文中，基于一种称为矩阵半张量积的新矩阵乘积的矩阵方法研究了基于k-外部稳定集 ( k- ESS)的图结构。通过定义图的节点子集的特征向量和特征值，分别以矩阵形式提出了图的k- ESS、最小k- ESS和k-核的三个充要条件。使用这些条件，k- ESS 矩阵、最小k- ESS 矩阵和k- 的概念引入核矩阵。这些矩阵提供了图的相应结构的完整信息。进一步设计了三种算法，通过进行一系列矩阵运算，分别找出图的这三种结构。最后通过实例验证了结果的正确性和有效性。所提出的方法和结果可能为研究网络系统领域中与图结构相关的问题提供一种新方法。