Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

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

doi:10.1016/S0024-3795(03)00582-2

Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

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

Research output: Contribution to journal › Conference article › peer-review

50 Citations (Scopus)

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Abstract

We characterize the possible lists of orderedmultiplicities among matrices whose graph is a generalized star (a tree in which atmost one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem (IEP) for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible orderedmultiplicities is equivalent to the IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.

Original language	English
Pages (from-to)	311-330
Number of pages	20
Journal	Linear Algebra and its Applications
Volume	373
Issue number	Suppl.
DOIs	https://doi.org/10.1016/S0024-3795(03)00582-2
Publication status	Published - 1 Nov 2003
Event	Conference on Combinatorial Matrix Theory - Pohang, Korea, Republic of Duration: 14 Jan 2002 → 17 Jan 2002

Keywords

Hermitian matrices
Eigenvalues
Multiplicities
Trees
Inverse eigenvalue problems

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title = "Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars",

abstract = "We characterize the possible lists of orderedmultiplicities among matrices whose graph is a generalized star (a tree in which atmost one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem (IEP) for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible orderedmultiplicities is equivalent to the IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.",

keywords = "Hermitian matrices, Eigenvalues, Multiplicities, Trees, Inverse eigenvalue problems",

author = "Johnson, {Charles R.} and Ant{\'o}nio Leal-Duarte and Saiago, {Carlos Manuel}",

note = "1- This research was supported by Centro de Matem{\'a}tica da Universidade de Coimbra. 2- Research supported in part by Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia, Portugal, through the research grant SFRH/BD/899/2000. Part of the research was done while visiting the College of William and Mary.; Conference on Combinatorial Matrix Theory ; Conference date: 14-01-2002 Through 17-01-2002",

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Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars. / Johnson, Charles R.; Leal-Duarte, António; Saiago, Carlos Manuel.
In: Linear Algebra and its Applications, Vol. 373, No. Suppl., 01.11.2003, p. 311-330.

Research output: Contribution to journal › Conference article › peer-review

TY - JOUR

T1 - Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

AU - Johnson, Charles R.

AU - Leal-Duarte, António

AU - Saiago, Carlos Manuel

N1 - 1- This research was supported by Centro de Matemática da Universidade de Coimbra. 2- Research supported in part by Fundação para a Ciência e a Tecnologia, Portugal, through the research grant SFRH/BD/899/2000. Part of the research was done while visiting the College of William and Mary.

PY - 2003/11/1

Y1 - 2003/11/1

N2 - We characterize the possible lists of orderedmultiplicities among matrices whose graph is a generalized star (a tree in which atmost one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem (IEP) for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible orderedmultiplicities is equivalent to the IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.

AB - We characterize the possible lists of orderedmultiplicities among matrices whose graph is a generalized star (a tree in which atmost one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem (IEP) for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible orderedmultiplicities is equivalent to the IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.

KW - Hermitian matrices

KW - Eigenvalues

KW - Multiplicities

KW - Trees

KW - Inverse eigenvalue problems

U2 - 10.1016/S0024-3795(03)00582-2

DO - 10.1016/S0024-3795(03)00582-2

M3 - Conference article

SN - 0024-3795

VL - 373

SP - 311

EP - 330

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

IS - Suppl.

T2 - Conference on Combinatorial Matrix Theory

Y2 - 14 January 2002 through 17 January 2002

ER -

Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

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Keywords

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