The structure of matrices with a maximum multiplicity eigenvalue

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

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


There is remarkable and distinctive structure among Hermitian matrices, whose graph is a given tree $T$ and that have an eigenvalue of multiplicity that is a maximum for $T$. Among such structure, we give several new results: (1) no vertex of $T$ may be ``neutral''; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest.
Original languageEnglish
Pages (from-to)875-886
Number of pages12
JournalLinear Algebra and its Applications
Issue number4
Publication statusPublished - 1 Aug 2008


  • Hermitian matrices
  • Eigenvalues
  • Multiplicities
  • Maximum multiplicity
  • Trees
  • Path cover number
  • Parter vertices


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