超图的距离谱与张量谱理论

The distance spectral theory of graphs has been studied extensively, which is a current research focus in spectral graph theory. The distance spectral theory of hypergraphs attracts attention only very recently. With progressive deepening for the study of tensors, the tensor spectral theory of hypergraphs has attracted more and more attention, and becomes a hot spot in spectral graph theory and multilinear algebra. In preliminary studies, the applicant found that there are some common features in the study of the distance and tensor spectral theories of hypergraphs, and obtained some preliminary results on the distance spectral radius and H-spectral radius of hypergraphs. In this project, we plan to study the distance spectra and tensor spectra of hypergraphs, by invesgating topics such as distance spectral radius, other distance eigenvalues, adjacency (H, Z)-spectral radius, signless Laplacian (H, Z)-spectral radius, we try to reveal to inner relationship between the structure and the spectral proerties of distance matrix, adjacency tensor, and signless adjacency tensor for hypergraphs. The progress of these problems belongs to the futher development of spectral graph theory, and has important theoretical and practical significance.

图的距离谱理论已经得到较为深入的研究，是图谱理论的一个研究热点，而超图的距离谱理论最近刚刚才引起注意。随着张量研究的深入，超图的张量谱理论目前已经引起很多学者的注意，并成为图谱理论与多重线性代数领域的一个研究热点。申请者在前期研究中发现超图的距离谱与超图的张量谱的研究有一些相互借鉴的地方，得到超树的距离谱半径及超图H-谱半径的一些初步结论。本课题拟研究超图的距离与张量谱，通过距离谱半径、其他距离特征值、邻接(H、Z)-谱半径、无符号Laplacian (H、Z)-谱半径等问题的探究，来揭示超图的结构性质与距离矩阵、邻接张量、无符号Laplacian张量的谱性质的内在联系。这些问题的研究属于图的谱理论的深入发展，具有重要的理论意义与实际价值。

本项目主要研究了超图的距离谱和张量谱的一些组合与极值性质。给定若干结构参数的情形下刻画了距离谱半径、距离Laplacian谱半径及距离无符号Laplacian谱半径取得最大值或最小值的图或超图，也讨论了其他距离特征值的性质。在张量谱方面，首先研究了线性超图的张量谱半径的紧的界，刻画了若干超图类中α-谱半径(包含谱半径和无符号Laplacian谱半径为特殊情况)取得最大值的一致超图和不一定一致的一般超图。这些结果丰富了超图的距离谱和张量谱理论，拓展了组合矩阵论的研究内容，可促进组合矩阵论及其相关数学领域的研究与应用，为理论计算机科学和大数据的研究提供可能的数学基础理论。