樹的特徵值譜的探討

The study of the spectrum for eigenvalues of trees

莊銘宏(Ming-Hung Chuang)

指導教授 ： 高金美

淡江大學/理學院/中等學校教師在職進修數學教學碩士學位班/碩士(2011年)

https://doi.org/10.6846/TKU.2011.00197

摘要

在圖G＝(V,E)中，由E中不共用頂點的邊所成的集合稱為G的一個匹配，含有邊數最多的匹配稱為G的最大匹配。一個最大匹配的個數稱為G的匹配數。令M是G的一個最大匹配。如果G的每個頂點都可以在M中找到一個以其為端點的邊，則稱M是G的一個完美匹配，此時2|M|=|V(G)|。如果存在唯一一個點無法在M中找到一個以其為端點的邊，則稱M是G的一個幾乎完美匹配，此時2|M|＋1=|V(G)|。一個不含迴路的連通圖，稱為樹。在本論文中我們探討具有幾乎完美匹配的所有樹的鄰接矩陣和Laplacian矩陣的最大特徵值的上界和下界，首先，我們找出8個具有幾乎完美匹配的樹，且其最大特徵值可能為最大者，經由計算它們的特徵多項式及最大特徵值，而得到了具有幾乎完美匹配的所有樹的最大特徵值的上界及下界。

關鍵字

匹配 ； 最大匹配 ； 完美匹配 ； 幾乎完美匹配

並列摘要

Let G=(V,E) be a graph. A set of pairwise vertex disjoint edges of G is called a matching of G. A matching of maximum cardinality is called a maximum matching of G. The cardinality of a maximum matching of G is called the matching number of G. Let M be a maximum matching of G. If every vertex of G is saturated by M, then we call M a perfect matching of G, i.e. 2|M|=|V(G)|. If there exists only one vertex not saturated by M, then we call M a nearly perfect matching of G, i.e. 2|M|＋1=|V(G)|.
In this thesis, we mainly discuss an upper bound and a lower bound of the largest adjacency eigenvalues and Laplacian eigenvalues of trees with nearly perfect matchings. First we find eight trees which may contain the maximum of the largest eigenvalue of all trees with a nearly perfect matching. We calculate the characteristic polynomials and eigenvalues of their adjacency matrices and Laplacian matrices separately. We get an upper bound and a lower bound of trees with nearly perfect matchings from there.

並列關鍵字

matching ； maximum matching ； perfect matching ； nearly perfect matching

參考文獻

[1] Y. Chen, Properties of spectra of graphs and line graphs, Appl. Math. J. Chinese Univ. Ser. B, 17(3):371-376, 2002.

[3] Jiming Guo, On the second largest Laplacian eigenvalue of trees, Linear Algebra Appl., 2005, 404：251-261.

[4] Yaoping Hou and Jiongsheng Li, Bounds on the largest eigenvalues of trees with a given size of matching, Linear Algebra Appl.，2002，342：203-217．

[7] Guanghui Xu, On the spectral radius of trees with perfect matchings, Combination and Graph Theory, World Scientific, Singapore, 1997.

參考文獻