通过图的最大团表示共正矩阵的零点
Representation of Zeros of a Copositive Matrix via Maximal Cliques of a Graph
AI总结：
本文研究了共正矩阵与图论的新联系，证明了共正矩阵的归一化零点集合可表示为极小零点子集凸包的并集，这些子集与基于极小零点构建的特殊图的最大团相关，并开发了构建归一化极小零点和所有归一化零点集合的算法。
摘要：
共正矩阵与图论之间存在深刻的联系。共正矩阵为构建和求解各种具有挑战性的图相关问题提供了强大的工具。反过来，图论提供了丰富的概念和技术，可用于分析共正矩阵的关键性质，包括其特征值和谱。在本文中，我们展示了共正矩阵与图论之间关系的新方面。聚焦于共正矩阵的归一化零点集合，我们研究了其性质，并证明了该集合可以表示为极小零点子集的凸包的并集。我们证明了这些子集与基于该矩阵的极小零点集合构建的特殊图的最大团集合相关联。我们开发了一种算法，用于构建共正矩阵的归一化极小零点集合和所有归一化零点集合。
英文摘要：
There is a profound connection between copositive matrices and graph theory. Copositive matrices provide a powerful tool for formulating and solving various challenging graph-related problems. Conversely, graph theory provides a rich set of concepts and techniques that can be applied to analyze key properties of copositive matrices, including their eigenvalues and spectra. In this paper, we present new aspects of the relationship between copositive matrices and graph theory. Focusing on the set of normalized zeros of a copositive matrix, we investigate its properties and demonstrate that this set can be expressed as a union of convex hulls of subsets of minimal zeros. We show that these subsets are connected with the set of maximal cliques of a special graph constructed on the basis of the set of minimal zeros of this matrix. We develop an algorithm for constructing both the set of normalized minimal zeros and the set of all normalized zeros of a copositive matrix.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
