Eigenvalues, multiplicities and graphs

Charles R. Johnson, António Leal-Duarte, Carlos Manuel Saiago, David A. Sher

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Abstract

For a given graph, there is a natural question of the possible lists of multiplicities for the eigenvalues among the spectra of Hermitian matrices with that graph (no constraint is placed upon the diagonal entries of the matrices by the graph). Here, we survey some of what is known about this question and include some new information about it. There is a natural focus upon the case in which the graph is a tree. In this event, there is remarkable structure to the possible lists. Both the general theory and a summary of specific results is given. At the end, this allows to give, in compact tabular form, all lists for trees on fewer than 11 vertices (a potentially valuable tool for further work). There is a brief discussion of non-trees.
Original languageEnglish
Title of host publicationAlgebra and Its Applications
EditorsDinh V. Huynh, S. K. Jain, S. R. López-Permouth
Place of PublicationProvidence, RI
PublisherAmerican Mathematical Society
Pages167-183
Number of pages17
Volume419
ISBN (Electronic)978-0-8218-8098-2
ISBN (Print)978-0-8218-3842-6
DOIs
Publication statusPublished - 2006

Publication series

NameContemporary Mathematics
PublisherAmerican Mathematical Society
Number419

Keywords

  • Eigenvalue
  • Graph
  • Hermitian matrix
  • Multiplicities
  • Parter vertex
  • Symmetric matrix
  • Tree

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