单双圈混合图的零维数
发布时间：2018-07-17 19:47
【摘要】：谱图理论是图论的重要分支,主要是研究图的谱性质与结构性质之间的关系,通过图的谱性质刻画图的结构性质.混合图是既含有向边又含无向边的图,是简单图和定向图的推广.混合图的Hermite谱理论是谱图理论近年来涌现的重要研究方向.混合图的零维数是指在混合图Hermite邻接谱中0特征值的重数,是混合图的重要谱参数,一直是图谱理论的研究热点之一.Mohar在[29]中确定了所有零维数为n-2的n阶混合图,利用它对Hermite邻接同谱图进行分类,并且构造了不能被Hermite谱确定的一类图.我们在[36]中刻画了所有零维数为n-3的n阶混合图,证明了所有零维数为n-3的n阶连通混合图能被Hermite谱确定.本文分别刻画了零维数为n-4、n-5、n-6的n阶单圈混合图,以及零维数为n-4、n-5的n阶双圈混合图.本文组织结构如下:第一章首先介绍谱图理论的研究背景,随后介绍了本文用到的一些概念与术语.最后介绍了本文的研究问题、研究进展以及所取得的主要结论.第二章先给出了混合图的零维数分解定理,利用该定理刻画了零维数为n-4、n-5、n-6的n阶单圈混合图.第三章刻画了零维数为n-4、n-5的n阶双圈混合图.
[Abstract]:Spectral graph theory is an important branch of graph theory. It mainly studies the relationship between spectral properties and structural properties of graphs, and characterizes the structural properties of graphs by spectral properties of graphs. A mixed graph is a graph with both directed and undirected edges. It is a generalization of simple graphs and directed graphs. Hermite spectrum theory of mixed graphs is an important research direction in recent years. The zero dimension of a mixed graph is the multiplicity of the zero eigenvalues in the Hermite adjacent spectrum of a mixed graph, which is an important spectral parameter of the mixed graph. It has always been one of the hotspots of the graph theory. Mohar has determined all n-order mixed graphs with zero dimension n-2 in [29]. It is used to classify Hermite adjacent isospectral graphs and construct a class of graphs which can not be determined by Hermite spectrum. We characterize all n-order mixed graphs with zero dimension n-3 in [36], and prove that all n-dimensional n-3 n-connected mixed graphs can be determined by Hermite spectrum. In this paper, we characterize n-order n-cycle mixed graphs with zero dimension n-4n -5n -6 and n-order bicyclic mixed graphs with zero dimension n-4 + n-5 respectively. The structure of this paper is as follows: the first chapter introduces the research background of spectrum theory, and then introduces some concepts and terms used in this paper. Finally, this paper introduces the research problems, research progress and main conclusions. In chapter 2, we first give the zero-dimensional decomposition theorem of mixed graphs. By using this theorem, we characterize n-4n-5n- (n-6) -order monocyclic mixed graphs with zero dimension. In chapter 3, we characterize n-order bicyclic mixed graphs with zero dimension n-4n -5.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5
本文编号：2130747
[Abstract]:Spectral graph theory is an important branch of graph theory. It mainly studies the relationship between spectral properties and structural properties of graphs, and characterizes the structural properties of graphs by spectral properties of graphs. A mixed graph is a graph with both directed and undirected edges. It is a generalization of simple graphs and directed graphs. Hermite spectrum theory of mixed graphs is an important research direction in recent years. The zero dimension of a mixed graph is the multiplicity of the zero eigenvalues in the Hermite adjacent spectrum of a mixed graph, which is an important spectral parameter of the mixed graph. It has always been one of the hotspots of the graph theory. Mohar has determined all n-order mixed graphs with zero dimension n-2 in [29]. It is used to classify Hermite adjacent isospectral graphs and construct a class of graphs which can not be determined by Hermite spectrum. We characterize all n-order mixed graphs with zero dimension n-3 in [36], and prove that all n-dimensional n-3 n-connected mixed graphs can be determined by Hermite spectrum. In this paper, we characterize n-order n-cycle mixed graphs with zero dimension n-4n -5n -6 and n-order bicyclic mixed graphs with zero dimension n-4 + n-5 respectively. The structure of this paper is as follows: the first chapter introduces the research background of spectrum theory, and then introduces some concepts and terms used in this paper. Finally, this paper introduces the research problems, research progress and main conclusions. In chapter 2, we first give the zero-dimensional decomposition theorem of mixed graphs. By using this theorem, we characterize n-4n-5n- (n-6) -order monocyclic mixed graphs with zero dimension. In chapter 3, we characterize n-order bicyclic mixed graphs with zero dimension n-4n -5.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5
【参考文献】
相关期刊论文 前2条
1 ;A Note on the Nullity of Unicyclic Graphs[J];数学研究与评论;2010年05期
2 范益政;关于混合图的特征向量的结构(英文)[J];黑龙江大学自然科学学报;2004年04期
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