# 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

## Access Full Article

top## Abstract

top## How to cite

topRosário 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

top- [1] R. Fernandes, On the inverse eigenvalue problems: the case of superstars. Electronic Journal of Linear Algebra 18 (2009), 442-461. Zbl1218.05033
- [2] R. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press New York (1985). Zbl0576.15001
- [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] 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] 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] 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] 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] 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] G.Wiener, Spectral multiplicity and splitting results for a class of qualitativematrices, Linear Algebra and its Applications 61 (1984), 15-18.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.