圖的拉普拉斯特徵值之探討

令G為一簡單圖且A(G)為圖G的鄰接矩陣，D(G)為圖G的度對角矩陣，其中對角線元素dii為圖G上的點vi的度數，即dii=deg(vi)。定義L(G)=D(G)–A(G)為圖G的拉普拉斯矩陣，此拉普拉斯矩陣的特徵值稱為圖G的拉普拉斯特徵值。已知任一圖的最小拉普拉斯特徵值必為零且其餘皆為正數，而其最大特徵值又稱為此圖的拉普拉斯譜半徑。設f(n)為拉普拉斯譜半徑恰等於點數n且所有特徵值皆為整數的連通圖之總個數，Fiedler證明了圖G的拉普拉斯特徵值為其點數若且唯若圖G的補圖是不連通的。在本論文中我們利用此性質推得f(n)的遞迴關係式，並證明之。

關鍵字

連通圖；鄰接矩陣；拉普拉斯矩陣；拉普拉斯特徵值；遞迴關係式

並列摘要

Let G be a simple graph and A(G) the adjacency matrix of G. Let D(G) be a diagonal matrix such that dii=deg(vi) where vi is the vertex of G. Define L(G)＝D(G)－A(G), we call that L(G) is the Laplacian matrix of G and the eigenvalues of L(G) is the Laplacian eigenvalues. Since all Laplacian eigenvalues of G are nonnegative numbers, the smallest one is 0. We call the largest Laplacian eigenvalue is the Laplacian radius of G. Let f(n) be the number of connected graphs with n vertices having n as its Laplacian radius and all Laplacian eigenvalues being integers. In this thesis we obtain a recursive relation for f(n) to calculate the number of those graphs.

並列關鍵字

Connected graph ； Laplacian matrix ； Laplacian Eignvalues ； Recursive relation

參考文獻

〔1〕J.M. Cuo, On the laplacian spectral radius of a tree, Linear Algebra Appl, 368:379-385,2003

〔3〕R. Grone, R. merris and V. S. Sunder, The laplacian spectrum of a Graph, SIAM J. Matrix Anal, Appl, 11:218-238,1990

〔4〕Sheng-biao Hu, The laplacian eigenvalues of graphs, Journal of Mathematic,Wuhan, 6:661-663,2007

〔5〕Torsten Sander, An introduction to graph eigenvalues and Eigenvectors, Version 18.2005

〔2〕D. Cvetkovic’, M. Dooband H.Sachs, Spectra of graph-theory and Application, Johann Ambrosius Barth Verlag, 1995

Google Scholar

國際替代計量

圖的拉普拉斯特徵值之探討

全文下載

主題瀏覽