Abstract
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.
Highlights
A graph is a figure consisting of nodes and edges; nodes represent things; an edge linking two nodes represents the relationship between the two things
We present the main results of the paper, including theoretic and algorithm results. e theoretic part contains matrix formulations of k-externally stable set (ESS), minimum k-externally stable set (k-ESS), and k-kernels of graphs. e algorithm part consists of algebraic algorithms of finding all the k-ESSes, minimum k-ESSes, and k-kernels of a graph
Further Results on k-Kernels. e k-kernels of a graph play a key role in describing the structures of graphs and have a wide application prospect in the network systems [29]
Summary
A graph is a figure consisting of nodes and edges; nodes represent things; an edge linking two nodes represents the relationship between the two things. An externally stable set (ESS) of a graph is a set of nodes that any node outside of the set is linked with a node in the set. A k-externally stable set (k-ESS) of a graph is an advanced version of an ESS that any node outside of the set is linked with a node in the set by a path of length k. An internally stable set of a graph is a set of nodes where any two nodes are not linked with each other. A k-kernel of a graph is an advanced form of a kernel, that is, a k-kernel of a graph is both a k-ESS and a k-internally stable set (k-ISS), where a k-ISS is a set of nodes where any two nodes are not linked with each other by a path of length k. ISS, ESS, kernel, k-ISS, k-ESS, and k-kernels are important modes of describing the structure of graphs
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