簡易檢索 / 詳目顯示
| 研究生： |
羅文隆 |
|---|---|
| 論文名稱： |
連通圖的拉普拉斯與無符號拉普拉斯 譜半徑之研究 On the Laplacian and the Signless Laplacian Spectral Radius of a Connected Graph |
| 指導教授： | 張宜武 |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2015 |
| 畢業學年度： | 103 |
| 語文別： | 英文 |
| 論文頁數： | 34 |
| 中文關鍵詞： | 圖 、鄰接矩陣 、拉普拉斯矩陣 、無符號拉普拉斯矩陣 、譜半徑 、拉普拉斯譜半徑 、無符號拉普拉斯譜半徑 |
| 外文關鍵詞： | grpah, adjacency matrix,, Laplacian matrix, signless Laplacian matrix, spectral radius, Laplacian spectral radius, signless Laplacian spectral radius |
| 相關次數： | 點閱：49 下載：11 |
| 分享至: |
| 查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報 |
圖的譜半徑在數學方面以及其他領域有非常多的應用。在這篇論文裡，我們整理有關連通圖的拉普拉斯與無符號拉普拉斯譜半徑的論文。本文一開始探討一些圖的譜理論，並找出這些界限的關係。然後，我們將討論更精確的圖之拉普拉斯與無符號拉普拉斯譜半徑。最後，我們給一個例子，並使用前面所探討過的性質分析之。
The spectral radius of a graph has been applied in mathenatics and in diverse disciplines.In this thesis, we survey some papers about the Laplacian spectral radius and the signless Laplacian spectral radius of a connected graph. Initially, we discuss some properties about the spectral graphs and find the relations between these bounds. Then, we discuss the upper bounds and lower bounds of the Laplacian and signless Laplacian spectral radius of a graph. In the end, we give an example and analyze it.
Contents
中文摘要 .................................................i
Abstract ...............................................ii
1 Introduction ..........................................1
2 Preliminaries .........................................3
2.1 Definitions and Notations ...........................3
2.2 Some Basics in Matrix Theory.........................6
2.3 Some Properties of Spectral Graphs ................. 8
3 Some Properties of the Spectral Radius of a Graph.....11
3.1 Introduction ...................................... 11
3.2 More Connections to Matrix Theory ................. 13
3.3 Some Relations Among r(G), l(G) and m(G) ...........14
3.4 More Discussions ...................................16
4 Main Results .........................................23
4.1 Sharp Upper Bounds For l(G) and m(G) ...............23
4.2 Sharp Lower Bounds For l(G) and m(G)................27
4.3 Examples........................................... 30
5 Conclusion............................................32
[1] Douglas B.West, Introduction to graph theory, Prentice Hall, 1996.
[2] A.M.Yu, M.Lu, F.Tian, On the spectral radius of graphs, Linear Algebra Appl, 387, 2004, 41-49.
[3] Yuan Hong, Xiao-Dong Zhang, Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees, Discrete Mathematics, 296, 2005, 187-197.
[4] Tomohiro Kawasaki, A sharp upper bound for the largest eigenvalue of the Laplacian matrix of a tree, Portland State University M.S. in Mathematical Sciences, 296, 2011, 187-197.
[5] N.Biggs, Algebraic graph theory, second ed, Cambridge University Press, Cambridge, 1995.
[6] Meyer, C. D. (Carl Dean), Matrix analysis and applied linear algebra, Society for Industrial and Applied, 2000.
[7] G. Chris, R. Golden, Algebraic graph theory, Springer-Verlag, New York, Inc, 2001.
[8] Q. Li, K. Feng, On the largest eigenvalue of a graph, Acta Math, Appl. Sinica 2 (in Chinese): 167-175, 1979 .
[9] J. Shu, Y. Hong, K. Wnren, A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph, Linear Algebra Appl, 347, 2002, 123-129 .
[10] Jianxi Li, Wai Chee Shiu, Wai Hong Chan, The aplacian spectral radius of some graphs, Linear Algebra Appl, 431, 2009, 99–103.
[11] Dragos M. Cvetkovic, Michael Doob, Horst Sachs, Spectra of graphs : theory and application , Academic Press, 1979.
[12] Cvetkovic D., Applications of Graph Spectra: An introduction to the literature, Applications
of Graph Spectra, Zbornik radova 13(21), ed. D.Cvetkovi c, I.Gutman, Mathematical Institute SANU, Belgrade, 2009, 7-31.
[13] Ji-Ming Guo, The effect on the Laplacian spectral radius of a graph by adding or grafting edges, Linear Algebra Appl, 413, 2006, 59–71.
[14] Lihua Feng, Qiao Li and Xiao-Dong Zhang, Some Sharp Upper Bounds on the Spectral Radius of Graphs, TAIWANESE JOURNAL OF MATHEMATICS, 2007.
[15] Bao-Xuan Zhu, On the signless Laplacian spectral radius of graphs with cut vertices, Linear Algebra Appl, 433, 2010, 928–933.
[16] JIAQI JIANG, Introduction To Spectral Graph Theory, 2012.
[17] Zdenek Dvorak, Bojan Mohar, Spectral radius of finite and infinite planar graphs and of graphs of bounded genus, Journal-ref: J. Combin. Theory Ser. B 100 (2010) 729-739, arXiv:
0907.1591.
[18] M. N. Ellingham, Xiaoya Zha, The spectral radius of graphs on surfaces, Journal of Combinatorial Theory, Series B, 78, 2000, 45–56.
[19] Xiao-Dong Zhang, The Laplacian eigenvalues of graphs: a survey, Linear Algebra Research Advances, Editor: Gerald D. Ling, pp. 201-228,2007, arXiv:1111.2897v1 .
