We consider an inverse problem for Schrödinger operators on connected equilateral graphs with standard matching conditions. We calculate the spectral determinant and prove that the asymptotic distribution of a subset of its zeros can be described by the roots of a polynomial. We verify that one of the roots is equal to the mean value of the potential and apply it to prove an Ambarzumian type result, i.e., if a specific part of the spectrum is the same as in the case of zero potential, then the potential has to be zero.

中文翻译：

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图上的谱行列式和 Ambarzumian 类型定理

我们在具有标准匹配条件的连通等边图上考虑 Schrödinger 算子的逆问题。我们计算谱行列式并证明其零点子集的渐近分布可以由多项式的根来描述。我们验证其中一个根等于电位的平均值，并将其应用于证明 Ambarzumian 类型的结果，即，如果频谱的特定部分与零电位的情况相同，则电位具有为零。