关于实对称阵的特征值重数的组合性质研究

In combinatorial matrix theory, the corn is the qualitative analysis of matrices, and it reveals the combinatorial properties of matrices. The relationship between matrices and their associated graphs is an example for the combination of numbers with shapes, which reflects the mathematical beauty of helping numbers by shapes and assisting shapes by numbers.. The concepts of P-vertex and P-set of real symmetric matrices are proposed based on the multiplicity of eigenvalues of matrices and Cauchy's Interlacing Theorem. This project will focus on the properties of P-vertex and P-set of real symmetric matrices, mainly including: the existence of P-vertex and the characterization of P-vertex of real symmetric matrices, the characterization of the associated graphs of real symmetric matrices with the maximal number of P-vertices, the “continuity” of the number of P-vertices of real symmetric matrices, the characterization of the associated graphs of real symmetric matrices containing a P-set of maximal size, the graph theoretical characterization of the relationship between P-vertices and P-set of real symmetric matrices, etc.. The aim of this project is to extend the research on the P-vertices and P-set of real symmetric matrices from acyclic matrices (the matrices whose associated graphs are trees) to cyclic matrices (the matrices whose associated graphs contain cycles), strengthen the current research techniques, develop some new research methods, generalize the known results, and try to apply the acquired results to other related mathematical problems, such as minimum rank, inverse eigenvalues, etc.

组合矩阵论的核心内容是对矩阵进行定性分析，揭示矩阵蕴含的组合性质。矩阵与其伴随图之间的对应关系是数形结合的一个典范，体现了以形助数和以数辅形的数学美。. 实对称阵的P-点与P-集是基于矩阵的特征值重数以及Cauchy插值定理(Interlacing Theorem)所提出的定义。本项目将重点研究实对称阵的P-点与P-集的性质特征，主要包括：实对称阵的P-点存在性及P-点的特征刻画、具有极大P-点个数的实对称阵的伴随图的特征刻画、实对称阵P-点个数的“连续性”、具有极大P-集的实对称阵的伴随图的特征刻画、实对称阵的P-点与P-集之间联系的图论刻画等。. 本项目旨在将实对称阵的P-点与P-集研究从树矩阵(即伴随图为树的矩阵)延伸到含圈矩阵(即伴随图含圈的矩阵)，深化现有研究技巧，开拓新的研究手段，推广已有结果，并将所得结果应用到其他相关数学问题，如最小秩、特征值反问题等。

组合矩阵论的核心内容是对矩阵进行定性分析，揭示矩阵蕴含的组合性质，即仅与矩阵零位模式有关的性质。矩阵与其伴随图之间的联系是数形结合的一个典范，体现了以形助数和以数辅形的数学美。. 实对称阵的P-点与P-集是基于矩阵的特征值重数以及Cauchy插值定理所提出的定义。本项目重点研究了实对称阵的P-点与P-集的相关性质，所得的主要结果包括：树矩阵的极大P-集所含元素个数的“连续性”、奇异树矩阵的P-点个数的“连续性”、具有最大零度的树矩阵的P-点个数、给定秩的树矩阵的P-点个数等。. 此外，作为延伸拓展，本项目还研究了与矩阵、多项式、图参数相关的几个课题，主要包括：非奇异树矩阵与及逆矩阵的零位模式、多项式能量的积分表达式、树的距离矩阵的行列式的递推关系式、取得最小ABC指数的树的结构特征刻画等。. 总体来说，本项目给出了若干个关于树矩阵的P-点与P-集的研究结果，基本实现了预定的研究目标，同时我们也给出了多个与矩阵、多项式、图参数有关的研究成果。截至目前，在本项目的资助下，项目组合共发表了25篇论文，全部被SCI收录，另有部分论文已被接收或者审稿中。更为重要的是，在研究过程中，我们发掘出新的研究技巧，推广已有结果，并尝试将所得结果应用到其他相关问题，如M矩阵的逆矩阵、多项式能量等，为下一阶段的研究打下了坚实基础。