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    Title: 圖的拉普拉斯特徵值之探討
    Other Titles: The study of the Laplacian Eignvalues of graphs
    Authors: 戴思平;Tai, Su-Ping
    Contributors: 淡江大學中等學校教師在職進修數學教學碩士學位班
    高金美;Kau, Chin-Mei
    Keywords: 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;Connected graph;Laplacian matrix;Laplacian Eignvalues;recursive relation
    Date: 2011
    Issue Date: 2011-12-28 18:13:45 (UTC+8)
    Abstract: 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。
    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.
    Appears in Collections:[數學學系暨研究所] 學位論文

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