The number of distinct eigenvalues for which an index decreases multiplicity

Charles R. Johnson, António Leal Duarte, Carlos M. Saiago

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish
Pages (from-to)8-23
Number of pages16
JournalLinear Algebra and Its Applications
Volume516
DOIs
Publication statusPublished - 1 Mar 2017

Keywords

  • Diameter
  • Diameter minimal trees
  • Downer branch mechanism
  • Downer vertex
  • Eigenvalue multiplicity
  • Tree

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