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 Title: On the Spectrum of Trees Authors: 許立成;Li-Cheng Hsu Contributors: 數學研究所 Keywords: Bethe樹;\$v\$-symmetric eigenvector;symmetric eigenvector;skew symmetric vector;symmetric vector;\$i\$-level subtree of Bkn;Bethe tree;\$i\$-level set Date: 2009-03-26 Issue Date: 2009-09-22 11:10:00 (UTC+8) Publisher: 國立中央大學圖書館 Abstract: 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}\$. Appears in Collections: [數學研究所] 博碩士論文

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