【作者】 张旭；

【导师】 卜长江；

【作者基本信息】 哈尔滨工程大学 ， 应用数学， 2014， 硕士

【摘要】 上世纪90年代初,Peter Rowlinson和M.N. Ellingham在各自的文章中独立提出了图的星补概念.设μ是图G的k > 0重特征值,X (?) V (G),若|X| = k且μ不是G — X的导出子图的特征值,则称X是μ在图G中的星集,G — X是μ在图G中的星补.星补理论在强正则图,最小特征值—2的图,和图特征值重数的研究中都有着重要的作用.2005年,祁力群教授给出张量特征值的概念.邵嘉裕教授在2012年给出了张量乘法的定义.近两年一些关于一致超图的邻接张量和拉普拉斯张量的研究陆续出现.张量的谱理论研究尚处于起步阶段,本文给出张量强正定性的概念,并对张量的Hadamard不等式进行研究.主要结果如下:(1)给出星补中的非常重要的定理-重构定理的一个新的证明;(2)将图的邻接矩阵的星补理论推广到一般实对称矩阵的星补理论;(3)用星补技术证明了一类特殊图的特征值重数的上界,并且刻画了达到上界时的图的结构性质;(4)给出树的特征值重数上界的相关结果,改进了已有相关结论,并对达到上界时的图进行了刻画;(5)给出单圈图的特征值重数上界的相关结果;(6)给出一般张量的乘积概念;(7)给出强正定张量的定义和Hadamard不等式的证明.更多还原

【Abstract】 The research on star complements originated independently in papers by Peter Rowlinson and M.N. Ellingham. If μ is an eigenvalue of G of multiplicity k,then a star set for μ in G is a subset X of V(G) such that |X| = k and the induced subgraph G - X does not have μ as an eigenvalue. The induced subgraph G - X is called a star complement for μ in G. The theory of star complement has been used to study graphs with least eigenvalue -2, structure properties of strongly regular graphs and the multiplicities of graph eigenvalues.In 2005, Liqun Qi gave the definition of the eigenvalue of tensor. In 2012, Jiayu Shao gave a definition of tensor product. In recent years, some papers about the adjacency tensor and Laplace tensor of hypergraph have appeared. The study of the spectral of tensor is at beginning,we consider the eigenvalue of tensor based on the strong positive tensor. The main results of this paper are as follows:(1) Give a new proof of the Reconstruction Theorem for star complement;(2) Generalize the definition of star complement with adjacency matrix to real symmetric matrix;(3) The upper bounds on eigenvalue multiplicity of a specific graph is given, and the limit case is characterized;(4) The upper bounds on eigenvalue multiplicity of trees are given, which generalized the related results, and the limit case is characterized;(5) The upper bounds on eigenvalue multiplicity of unicyclic graphs are given;(6) Generalize the definition of product of tensors;(7) The definition and Hadamard inequation of strong positive definite tensor is obtained.更多还原