【摘要】：譜圖理論是圖論的重要分支,主要是研究圖的譜性質(zhì)與結(jié)構(gòu)性質(zhì)之間的關(guān)系,通過圖的譜性質(zhì)刻畫圖的結(jié)構(gòu)性質(zhì).混合圖是既含有向邊又含無向邊的圖,是簡單圖和定向圖的推廣.混合圖的Hermite譜理論是譜圖理論近年來涌現(xiàn)的重要研究方向.混合圖的零維數(shù)是指在混合圖Hermite鄰接譜中0特征值的重?cái)?shù),是混合圖的重要譜參數(shù),一直是圖譜理論的研究熱點(diǎn)之一.Mohar在[29]中確定了所有零維數(shù)為n-2的n階混合圖,利用它對Hermite鄰接同譜圖進(jìn)行分類,并且構(gòu)造了不能被Hermite譜確定的一類圖.我們在[36]中刻畫了所有零維數(shù)為n-3的n階混合圖,證明了所有零維數(shù)為n-3的n階連通混合圖能被Hermite譜確定.本文分別刻畫了零維數(shù)為n-4、n-5、n-6的n階單圈混合圖,以及零維數(shù)為n-4、n-5的n階雙圈混合圖.本文組織結(jié)構(gòu)如下:第一章首先介紹譜圖理論的研究背景,隨后介紹了本文用到的一些概念與術(shù)語.最后介紹了本文的研究問題、研究進(jìn)展以及所取得的主要結(jié)論.第二章先給出了混合圖的零維數(shù)分解定理,利用該定理刻畫了零維數(shù)為n-4、n-5、n-6的n階單圈混合圖.第三章刻畫了零維數(shù)為n-4、n-5的n階雙圈混合圖.
[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.
【學(xué)位授予單位】：安徽大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O157.5

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本文編號：2130747