TY - JOUR

T1 - The number of distinct eigenvalues for which an index decreases multiplicity

AU - Johnson, Charles R.

AU - Duarte, António Leal

AU - Saiago, Carlos M.

N1 - Research partially supported by the Centro de Matematica da Universidade de Coimbra (CMUC), University of Coimbra and Fundacao para a Ciencia e a Tecnologia, through European program COMPETE/FEDER (under the contract PTDC/MAT/112273/2009).
This work was carried out within the activities of CMA/FCT/UNL and it was partially supported by Fundacao para a Ciencia e a Tecnologia through the project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes).

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

KW - Diameter

KW - Diameter minimal trees

KW - Downer branch mechanism

KW - Downer vertex

KW - Eigenvalue multiplicity

KW - Tree

UR - http://www.scopus.com/inward/record.url?scp=84997820962&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2016.11.033

DO - 10.1016/j.laa.2016.11.033

M3 - Article

AN - SCOPUS:84997820962

VL - 516

SP - 8

EP - 23

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -