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題名 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作者 羅文隆 貢獻者 張宜武
羅文隆關鍵詞 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
grpah
adjacency matrix,
Laplacian matrix
signless Laplacian matrix
spectral radius
Laplacian spectral radius
signless Laplacian spectral radius日期 2015 上傳時間 24-Aug-2015 09:55:36 (UTC+8) 摘要 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.參考文獻 [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, Applicationsof 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 . 描述 碩士
國立政治大學
應用數學研究所
100751007資料來源 http://thesis.lib.nccu.edu.tw/record/#G0100751007 資料類型 thesis dc.contributor.advisor 張宜武 zh_TW dc.contributor.author (Authors) 羅文隆 zh_TW dc.creator (作者) 羅文隆 zh_TW dc.date (日期) 2015 en_US dc.date.accessioned 24-Aug-2015 09:55:36 (UTC+8) - dc.date.available 24-Aug-2015 09:55:36 (UTC+8) - dc.date.issued (上傳時間) 24-Aug-2015 09:55:36 (UTC+8) - dc.identifier (Other Identifiers) G0100751007 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/77867 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學研究所 zh_TW dc.description (描述) 100751007 zh_TW dc.description.abstract (摘要) 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。 zh_TW dc.description.abstract (摘要) 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. en_US dc.description.tableofcontents Contents中文摘要 .................................................iAbstract ...............................................ii1 Introduction ..........................................12 Preliminaries .........................................32.1 Definitions and Notations ...........................32.2 Some Basics in Matrix Theory.........................62.3 Some Properties of Spectral Graphs ................. 83 Some Properties of the Spectral Radius of a Graph.....113.1 Introduction ...................................... 113.2 More Connections to Matrix Theory ................. 133.3 Some Relations Among r(G), l(G) and m(G) ...........143.4 More Discussions ...................................164 Main Results .........................................234.1 Sharp Upper Bounds For l(G) and m(G) ...............234.2 Sharp Lower Bounds For l(G) and m(G)................274.3 Examples........................................... 305 Conclusion............................................32 zh_TW dc.format.extent 633303 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0100751007 en_US dc.subject (關鍵詞) 圖 zh_TW dc.subject (關鍵詞) 鄰接矩陣 zh_TW dc.subject (關鍵詞) 拉普拉斯矩陣 zh_TW dc.subject (關鍵詞) 無符號拉普拉斯矩陣 zh_TW dc.subject (關鍵詞) 譜半徑 zh_TW dc.subject (關鍵詞) 拉普拉斯譜半徑 zh_TW dc.subject (關鍵詞) 無符號拉普拉斯譜半徑 zh_TW dc.subject (關鍵詞) grpah en_US dc.subject (關鍵詞) adjacency matrix, en_US dc.subject (關鍵詞) Laplacian matrix en_US dc.subject (關鍵詞) signless Laplacian matrix en_US dc.subject (關鍵詞) spectral radius en_US dc.subject (關鍵詞) Laplacian spectral radius en_US dc.subject (關鍵詞) signless Laplacian spectral radius en_US dc.title (題名) 連通圖的拉普拉斯與無符號拉普拉斯 譜半徑之研究 zh_TW dc.title (題名) On the Laplacian and the Signless Laplacian Spectral Radius of a Connected Graph en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) [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, Applicationsof 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 . zh_TW
