图的逆特征值问题中的解放集
The liberation set in the inverse eigenvalue problem of a graph
AI总结：
本文改进 Matrix Liberation Lemma 的等价表述，降低应用难度，并将其用于直和矩阵，揭示其与图笛卡尔积零强制博弈的联系，解决若干六点图逆特征值问题的未决情形。
摘要：
图 $G$ 的逆特征值问题，是刻画其非对角元素模式由 $G$ 的邻接关系规定的实对称矩阵的所有特征值列表的问题。强谱性质是该问题中的一个有力工具，它识别出那些可在控制模式并保持特征值的同时对元素进行扰动的矩阵。Barrett 等人于 2020 年提出的 Matrix Liberation Lemma（矩阵解放引理）将这一概念推进到更一般的情形。本文重新考察 Matrix Liberation Lemma，并证明了一个等价命题，减少了应用该结果时的部分技术困难。我们在形如 $M=A \oplus B$ 的矩阵上测试该方法，并展示这一新途径如何补充仅依靠强谱性质所能获得的结果。在将该概念推广到图的直和时，我们发现了它与图的笛卡尔积上的零强制博弈之间令人惊讶的联系。全文中，我们应用这些结果解决了六个顶点上图的逆特征值问题的若干未决情形。
英文摘要：
The inverse eigenvalue problem of a graph $G$ is the problem of characterizing all lists of eigenvalues of real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of $G$. The strong spectral property is a powerful tool in this problem, which identifies matrices whose entries can be perturbed while controlling the pattern and preserving the eigenvalues. The Matrix Liberation Lemma introduced by Barrett et al.~in 2020 advances the notion to a more general setting. In this paper we revisit the Matrix Liberation Lemma and prove an equivalent statement, that reduces some of the technical difficulties in applying the result. We test our method on matrices of the form $M=A \oplus B$ and show how this new approach supplements the results that can be obtained from the strong spectral property only. While extending this notion to the direct sums of graphs, we discover a surprising connection with the zero forcing game on Cartesian products of graphs. Throughout the paper we apply our results to resolve a selection of open cases for the inverse eigenvalue problem of a graph on six vertices.
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