一致超图若干特征值和特征向量理论与应用研究

The eigenvalue and eigenvector problem of a hypergraph is an active and important research field in hypergraph theory. And it has extensive applications in the fields of computer science, communication networks, information science, statistical mechanics, quantum chemistry and medical imaging technology, etc. Especially, from the year of 2005 when Dr. Lek-Heng Lim from the USA innovatively pointed out that it is available to study the hypergraph eigenvalue theory by the tensor eigenvalue theory, then a lot of experts and scholars on graph theory and tensor theory began to pay close attention to this problem and had published dozens of related articles.. In the eigenvalue and eigenvector theory of hypergraphs, there are various tensors that are naturally associated with a hypergraph, such as the adjacency tensor, Laplacian tensor, signless Laplacian tensor. In this project, we further study the relevant theoretical properties and practical applications on the H-eigenvalues and H-eigenvectors of the adjacency tensor, Laplacian tensor and signless Laplacian tensor for a uniform hypergraph. Especially, the largest and the smallest H-eigenvalues with their eigenvectors of the above three tensors for a uniform hypergraph are researched, and their connections with bipartition, planarity, cospectrum, clique number, independence number and chromatic number of a uniform hypergraph are discussed. Also, we begin to study the theory and application on eigenvalues and eigenvectors of uniform directed hypergraphs.

超图的特征值和特征向量问题是超图理论的一个非常活跃而又重要的研究领域。而且它在计算机科学、通信网络、信息科学、统计力学、量子化学以及医学成像技术等中均有广泛的应用前景。特别是从2005年美国林力行博士创新性指出可用张量的特征值理论来研究超图的特征值理论以来，很多图论界和张量界专家学者开始关注此问题并陆续发表了几十篇相关文章。在超图的特征值和特征向量理论中，有各种张量能较自然地对应超图，比如邻接张量、拉普拉斯张量和无符号拉普拉斯张量。本项目中，将进一步研究一致超图的邻接张量、拉普拉斯张量和无符号拉普拉斯张量对应的H特征值及H特征向量的相关理论性质和实际应用问题。特别地，研究一致超图这三种张量最大与最小H特征值及其特征向量，并进一步讨论这些特征值及其特征向量与一致超图的二部划分、平面性、同谱性、团数、独立数和着色数等相关图论参数的应用关系。另外着手研究有向一致超图的特征值和特征向量理论与应用。

超图的特征值和特征向量问题是超图理论的一个非常活跃而又重要的研究领域。而且它在计算机科学、通信网络、信息科学、统计力学、量子化学以及医学成像技术等中均有广泛的应用前景。本项目中，进一步研究一致超图的邻接张量、拉普拉斯张量和无符号拉普拉斯张量对应的H特征值及H特征向量的相关理论性质和实际应用问题。特别地，研究了完全4一致超图邻接张量的特征值问题。另外着手研究有向一致超图的特征值和特征向量理论与应用。