阈值图允许少数不同特征值：一种新方法
Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach
- Ontario Tech University（安大略科技大学）
- University of Regina（里贾纳大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种新方法，证明任意阈值图的最小不同特征值数不超过4，并进一步证明所有连通阈值图都存在具有任意四个不同特征值的矩阵。
AI中文摘要：
对于任意图 $G$，我们关联一族实对称矩阵 $S(G)$，其中对于任意 $A \in S(G)$，$A$ 的非零非对角元的位置由 $G$ 的邻接结构决定。令 $q(G)$ 表示 $S(G)$ 中所有矩阵上不同特征值的最小数目。在这项工作中，我们提供了一种替代技术来证明：对于任意阈值图 $G$，$q(G) \leq 4$，如文献 [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53] 所述。此外，我们证明所有连通阈值图都允许一个具有任意四个不同特征值的矩阵。
英文摘要：
For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we provide an alternative technique to establish that $q(G) \leq 4$ for any threshold graph $G$ as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further
