Shaun M. Fallat , H. Tracy Hall , Rupert H. Levene , Seth A. Meyer , Shahla Nasserasr , Polona Oblak , Helena Šmigoc
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引用次数: 0
Abstract
Given a graph G, consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted. In particular, we construct trees with multiplicity lists that require a unique spectrum, up to shifting and scaling. This represents the most extreme possible failure of spectral arbitrariness for a multiplicity list, and greatly extends all previously known instances of this phenomenon, in which only single linear constraints on the eigenvalues were observed.
给定一个图 G，考虑实对称矩阵族，其非零对角线项的模式对应于 G 的边。通过组合方法，我们确定了一系列图形和多重性列表，它们的可实现光谱受到了很大限制。特别是，我们构建的树与多重性列表需要唯一的频谱，直至移位和缩放。这代表了多重性列表频谱任意性可能出现的最极端故障，并大大扩展了之前已知的所有这种现象的实例，在这些实例中，只观察到对特征值的单一线性约束。
期刊介绍：
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.
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