本文選題：無符號拉普拉斯矩陣　切入點：特征方程　出處：《華東理工大學》2017年碩士論文　論文類型：學位論文

【摘要】：圖的譜理論作為圖論和組合矩陣理論的一個組成部分,已經(jīng)得到了越來越多研究者的關注,并且在量子化學、統(tǒng)計力學、計算機科學、通信網(wǎng)絡以及信息科學中均有著廣泛的應用.近年來,對于圖的特征值的研究也是十分活躍的課題.大量文章集中于對圖的無符號拉普拉斯矩陣、拉普拉斯矩陣以及鄰接矩陣相關性質的的研究,比如對于無符號拉普拉斯矩陣的特征值的研究,最大特征值以及最小特征值的界的研究以及擾動下的無符號拉普拉斯矩陣最小特征值的一些性質。在由E.R.van Dam,W.H Haemers 2003年寫的《Which graphs are determined by their spectrum》一書中指出在區(qū)分非同構圖中無符號拉普拉斯矩陣的譜比拉普拉斯矩陣的譜或者鄰接矩陣的譜能更好的反應圖的性質,同時圖的無符號拉普拉斯矩陣的最小特征值是圖的二部性的一個重要判斷指標。文獻[9]給出了無符號拉普拉斯矩陣的最小特征值達到最小時的極圖,即三角形在其某一端點處懸掛一條路.本篇論文主要對無符號拉普拉斯矩陣的最小特征值進行了研究.給出了在所有的非二部單圈圖中,無符號拉普拉斯矩陣的最小特征值達到最大的極圖。主要是利用移接變形,二次型,特征方程等方法對無符號拉普拉斯譜的一些結論更深層次的探討。
[Abstract]:As a component of graph theory and combinatorial matrix theory, the spectral theory of graphs has been paid more and more attention by researchers, and has been studied in quantum chemistry, statistical mechanics, computer science. In recent years, the research on the eigenvalues of graphs is also a very active subject. A large number of papers focus on the unsigned Laplace matrix of graphs. Studies on the related properties of Laplace matrices and adjacent matrices, such as the eigenvalues of unsigned Laplace matrices, The study of the bounds of maximum eigenvalue and minimum eigenvalue and some properties of minimum eigenvalue of unsigned Laplacian matrix under perturbation. In the book "Which graphs are determined by their spectrum", written by E.R. van DamW.H Haemers in 2003, it is pointed out in distinguishing nonisomorphism. The spectrum of the unsigned Laplacian matrix is better than the spectrum of the Laplace matrix or the spectrum of the adjacent matrix. At the same time, the minimum eigenvalue of the unsigned Laplacian matrix of a graph is an important criterion for the biparity of the graph. In reference [9], the pole graph of the unsigned Laplacian matrix when the minimum eigenvalue of the unsigned Laplacian matrix reaches the minimum is given. In this paper, the minimum eigenvalues of the unsigned Laplace matrix are studied. In all non-bipartite unicyclic graphs, the minimum eigenvalues of the unsigned Laplace matrix are studied. The minimum eigenvalue of unsigned Laplacian matrix reaches the maximum pole graph. Some conclusions of unsigned Laplacian spectrum are discussed deeply by means of shift deformation, quadratic form, characteristic equation and so on.
【學位授予單位】：華東理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O157.5

【參考文獻】

相關期刊論文 前2條

1 Rui-fang LIU;Hui-cai JIA;Jin-long SHU;;An Edge-rotating Theorem on the Least Eigenvalue of Graphs[J];Acta Mathematicae Applicatae Sinica;2015年04期

2 徐光輝,徐群芳,王勝奎;單圈圖最小特征值的Sharp下界[J];寧波大學學報(理工版);2003年03期

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本文編號：1592756