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

• Autorzy: Rosário Fernandes
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Eigenvalue multiplicities; Symmetric matrices; Trees; Two largest multiplicities

Kategorie tematyczne

• 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
• 05C38: Paths and cycles
• 15A18: Eigenvalues, singular values, and eigenvectors

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2014-09-10

zaakceptowano

2014-12-04

online

2015-01-12

Twórcy

autor

Rosário Fernandes

• Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade
  Nova de Lisboa, 2829-516 Caparica, Portugal

Bibliografia

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

Identyfikatory

DOI

10.1515/spma-2015-0001