本篇論文將研究圖上Sturm-Liouville算子 Ambarzumyan
問題，Ambarzumyan定理說明在 [0,π] 區間的古 Sturm-Liouville 問題，若 Neumann 邊值問題形成的集合為{n^2: n ∈ N ⋃ {0} }，則勢函數 q 恆為0。Pivovarchik在2004年證明了2個有關三邊等長星狀圖上的Sturm-Liouville 算子。之後，伍懋靈在他的論文裡研究邊長不相等的情況下有關Sturm-Liouville算子的 Ambarzumyan問題。在本論文中，我們將在4邊星狀圖和樹狀圖上對Neumann 邊值問題和 Dirichlet 邊值問題求它們的解Ambarzumyan 問題。最後我們發現可以將整個特徵問題的譜集合分割，每一類別都能形成一個 Ambarzumyan 問題的解。舉例來說，在邊長為 a, a, 2a, 2a 的4星狀圖上就存在3個Neumann邊值問題的Ambarzumyan問題的解。

We study the Ambarzumyan problem for Sturm-Liouville operator defined on graph. The classical Ambarzumyan Theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator defined on
the interval [0,π] are exactly {n^2: n ∈ N ⋃ {0} }, then the potential q=0. In 2005, Pivovarchik proved two similar theorems with uniform lengths a for the Sturm-Liouville operator defined on a 3-star graphs. Then Wu considered the Ambarzumyan problem for graphs
of nonuniform length in his thesis. In this thesis, we shall study the Ambarzumyan problem on more complicated trees, namely, 4-star graphs and caterpillar graphs with edges of different lengths. We
manage to solve the Ambarzumyan problem for both Neumann eigenvalues and Dirichlet eigenvalues. In particular, the whole spectrum can be partitioned into several parts. Each part forms the solution to one
Ambarzumyan problem. For example, for a 4-star graphs with edge lengths a, a, 2a, 2a form the solution to 3 different Ambarzumyan problems for the Neumann eigenvalues.

1. Introduction
2. Direct Problems
2.1 Four star-shaped graphs
2.2 caterpillar graphs
3. Inverse Problems
3.1 Four star-shaped graphs
3.2 caterpillar graphs
Appendix A. Figures of graphs
Appendix B. Tables for the solutions
of Ambarzumyan Problems.
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