The structure of matrices with a maximum multiplicity eigenvalue

• Autor(a) principal: Johnson, Charles R.
• Data de Publicação: 2008
• Outros Autores: Leal-Duarte, António, Saiago, Carlos Manuel
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10362/58385
• topic: Hermitian matrices; Eigenvalues; Multiplicities; Maximum multiplicity; Trees; Path cover number; Parter vertices

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.