| Summary: | 碩士 === 國立中央大學 === 數學研究所 === 97 === 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 $T=K_{1,Delta}$.
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