# The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

Special Matrices (2015)

• Volume: 3, Issue: 1, page 1-17, electronic only
• ISSN: 2300-7451

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## Abstract

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The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial algorithm, cases of equality are computed for M2.

## How to cite

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Rosário Fernandes. "The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree." Special Matrices 3.1 (2015): 1-17, electronic only. <http://eudml.org/doc/268700>.

@article{RosárioFernandes2015,
abstract = {The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial algorithm, cases of equality are computed for M2.},
author = {Rosário Fernandes},
journal = {Special Matrices},
keywords = {Eigenvalue multiplicities; Symmetric matrices; Trees; Two largest multiplicities; eigenvalue multiplicities; symmetric matrices; trees; two largest multiplicities; Hermitian matrix; combinatorial algorithm},
language = {eng},
number = {1},
pages = {1-17, electronic only},
title = {The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree},
url = {http://eudml.org/doc/268700},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Rosário Fernandes
TI - The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree
JO - Special Matrices
PY - 2015
VL - 3
IS - 1
SP - 1
EP - 17, electronic only
AB - The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial algorithm, cases of equality are computed for M2.
LA - eng
KW - Eigenvalue multiplicities; Symmetric matrices; Trees; Two largest multiplicities; eigenvalue multiplicities; symmetric matrices; trees; two largest multiplicities; Hermitian matrix; combinatorial algorithm
UR - http://eudml.org/doc/268700
ER -

## References

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1. [1] R. Fernandes, On the inverse eigenvalue problems: the case of superstars. Electronic Journal of Linear Algebra 18 (2009), 442-461. Zbl1218.05033
2. [2] R. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press New York (1985). Zbl0576.15001
3. [3] C.R. Johnson and A.Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999), 139-144. Zbl0929.15005
4. [4] C.R. Johnson and A.Leal Duarte, On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree, Linear Algebra and Applications 248 (2002), 7-21. Zbl1001.15004
5. [5] C.R. Johnson, A.Leal Duarte and C.M. Saiago, The Parter-Wiener theorem: refinement and generalization, SIAM Journal on Matrix Analysis and Applications 25 (2) (2003), 352-361. [Crossref] Zbl1067.15003
6. [6] C.R. Johnson, A.Leal Duarte and C.M. Saiago, 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, Linear Algebra and its Applications 373 (2003), 311-330. Zbl1035.15010
7. [7] C.R. Johnson, C. Jordan-Squire and D.A. Sher, Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree, Discrete Applied Mathematics 158 (2010), 681-691. [WoS] Zbl1225.05166
8. [8] S. Parter, On the eigenvalues and eigenvectors of a class of matrices, Journal of the Society for Industrial and Applied Mathematics 8 (1960), 376-388. Zbl0115.24804
9. [9] G.Wiener, Spectral multiplicity and splitting results for a class of qualitativematrices, Linear Algebra and its Applications 61 (1984), 15-18.

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