The number of distinct eigenvalues for which an index decreases multiplicity

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• 作者：Charles R. Johnson^a ; ^{crjohn@wm.edu} ; Antó ; nio Leal Duarte^b ; ^{leal@mat.uc.pt} ; Carlos M. Saiago^c ; ¹ ; ^{cls@fct.unl.pt}
• 关键词：15A18 ; O5CO5 ; O5C50
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：516
• 期：Complete
• 页码：8-23
• 全文大小：356 K
• 卷排序：516

文摘

For an Hermitian matrix A whose graph is a tree T, we study the number of eigenvalues of A whose multiplicity decreases when a particular vertex is deleted from T. Explicit results are given when that number of eigenvalues is less than 4 and an inductive result thereafter. The work is based, in part, on classical results about multiplicities, but also on some new facts, including a useful identity. This allows us to give strong bounds based on simple facts about the location of the vertex in the tree. Some facts about matrices whose graphs are not trees are included, and the classical diameter bound about the number of distinct eigenvalues for a tree follows.