We study an inverse eigenvalue problem (IEP) of reconstructing a special kind of symmetric acyclic matrices whose graph is a generalized star graph. The problem involves the reconstruction of a matrix by the minimum and maximum eigenvalues of each of its leading principal submatrices. To solve the problem, we use the recurrence relation of characteristic polynomials among leading principal minors. The necessary and sufficient conditions for the solvability of the problem are derived. Finally, a numerical algorithm and some examples are given.

中文翻译：

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一类特殊无环矩阵的特征值反问题

我们研究了重构一种特殊类型的对称非循环矩阵的特征值反问题 (IEP)，其图是广义星图。该问题涉及通过矩阵的每个主要主子矩阵的最小和最大特征值重建矩阵。为了解决这个问题，我们使用了领先的主要未成年人之间的特征多项式的递推关系。推导出问题可解的充要条件。最后给出了一个数值算法和一些例子。