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研究生： 許立成
Li-Cheng Hsu
論文名稱：
On the Spectrum of Trees
指導教授： 黃華民
Hua-Min Huang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 97
語文別： 英文
論文頁數： 26
中文關鍵詞： Bethe樹
外文關鍵詞： $v$-symmetric eigenvector, symmetric eigenvector, skew symmetric vector, symmetric vector, $i$-level subtree of Bkn, Bethe tree, $i$-level set
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    • 在1984年，Godsil 定義了 Bethe樹圖B(k,n)，並求出其譜半徑
    ho的上界滿足 $rho<2sqrt{k}$。在我們這篇論文中，我們找出Bethe樹圖的譜，利用此結論，我們又證明了任一樹圖T 的譜半徑滿足
    $$sqrt{Delta}leq
    ho< min{2sqrt{Delta-1}cos{(frac{pi}{D+2})},2sqrt{Delta}cos{(frac{pi}{r+2})}},$$
    其中D，r，Delta分別為此樹圖的直徑，半徑，與最大度數。此下界等號成立只發生在當T為完全二部圖K_{1,Delta}時。

    In 1984, Godsil defined the Bethe tree $B(k,n)$ and showed the spectral radius $
    ho$ of $B(k,n)$ satisfies $
    ho<2sqrt{k}$.
    In this thesis, we find the spectrum of $B(k,n)$. With this spectrum, we also show the spectral radius $
    ho$ of a tree $T$ satisfies
    $$sqrt{Delta}leq
    ho< min{2sqrt{Delta-1}cos{(frac{pi}{D+2})},2sqrt{Delta}cos{(frac{pi}{r+2})}},$$
    where $D$,$r$,$Delta$ are the diameter, radius, and the maximum degree of $T$ respectively. The equality of lower bound holds only when $T=K_{1,Delta}$.

    Contents 1 Historical Review and Overview of this thesis 1 1.1 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Basic Definitions of Graph Theory 4 3 Some Preliminaries on Eigenvalues of Graphs 7 3.1 The eigenvalues of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Equitable partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 The Spectrum of the Bethe Tree B(k, n) 12 4.1 Basic definitions and properties of Bethe tree . . . . . . . . . . . . . . . . . . 12 4.1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.1.2 Spectrum of B(1, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Equitable partition and symmetric eigenvector of B(k, n) . . . . . . . . . . . 16 4.2.1 An equitable partition of B(k, n) . . . . . . . . . . . . . . . . . . . . . 16 4.2.2 Symmetric eigenvectors of B(k, n) . . . . . . . . . . . . . . . . . . . . 18 4.3 v-symmetric eigenvector of B(k, n) . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Spectrum of B(k, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Spectral radius of Trees 22 5.1 Lower bound of the spectral radius of trees . . . . . . . . . . . . . . . . . . . 22 5.2 Upper bound of the spectral radius of trees . . . . . . . . . . . . . . . . . . . 24 5.3 Possible future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Bibliography 26

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    [6] C.D.Godsil, Spectra of trees, Ann. Disc. Math. 20(1984) 151-159.
    [7] Y. Hong, J.-L. Shu, K. Fang, A sharp upper bound of the spectral radius of graphs, J.
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    [9] S. Hu, The largest eigenvalue of unicyclic graphs, Discrete Mathematics 307(2007) 280-
    284.
    [10] D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ (2001).

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