超图谱理论中若干问题的研究

The main goal of this project is the research on the structure properties of uniform hypergraphs or directed uniform hypergraphs in virtue of the spectrum of hypergraphs, which are based on tensors eigenvalues theory. Some applications of the spectrum of hypergraphs on hypergraph partitioning will be also studied. Analytic methods, nonlinear optimization methods, tensor theory, some combinatorial methods concerning structure of uniform hypergraphs will be comprehensively used in this project. The main content of this proposal contains the following. Firstly, the extremal problems of spectrum will be studied by determining the extremal value of Laplace spectral radii of hypertrees, or of general hypergraphs, considering the extremal spectrum problems of hypergraphs with given chromatic number, or with given edge number. Secondly, to generalize some properties of several tensors of hypergraphs, the definition of the gereralized Laplace tensor of uniform hypergraph will be raised and studied. Thirdly, the relationship between the spectrum of the normalized Laplace tensor and the structure properties of uniform hypergraphs will be studied; we also explore the applications of normalized Laplace spectrum, or its H-eigenvalues and Z-eigenvalues in the partitioning problems of uniform hypergraphs.

本项目拟借助张量谱理论开展一致超图或有向一致超图谱问题的研究,同时开展超图谱在超图划分中的应用研究,拟综合运用分析不等式、非线性优化、张量代数以及超图结构分析等方法开展研究工作。具体研究内容如下(1)超图谱极值问题：超树与一般超图的Laplace谱半径的极值与极图刻画问题,色数或边数固定超图邻接谱半径的极值问题;(2)提出并研究超图以及有向超图的广义Laplace张量谱,试图从形式上统一几类超图张量谱的一些结论,进一步借此开展一般张量谱的性质研究;(3)超图的normalized Laplace谱问题：研究超图的normalized Laplace谱性质与超图的其他结构性质之间的关系,重点研究normalized Laplace谱或其H-谱、Z-谱在超图划分问题中的应用。

本项目的核心是基于张量谱理论开展一致超图或符号一致超图谱问题的研究,同时开展超图谱在超图划分中的应用研究。课题执行期间我们重点进行了超图谱极值问题的研究，刻画了最大度固定的一致超树的极值问题，同时刻画了赋权一致超图的谱极值问题；开展了符号超图的邻接谱和Laplace谱的研究；同时探讨了超图的均衡划分与谱半径的不等式关系； 充分使用张量特征值理论, 得到了谱半径的一些上、下界, 得到了邻接谱半径与超图的其他参数之间的关系。项目执行期间所取得的理论成果丰富了超图谱理论。.项目执行期间我们积极进行学术交流，多次参加学会的线上或线下学术交流会，同时邀请多位知名教授为课题的进展提供咨询和指导，开展多次学术讲座。项目执行期间课题组共培养8名研究生。