Eigenvalues, multiplicities and graphs /

This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues.

Détails bibliographiques
Cote:Libro Electrónico
Auteurs principaux: Johnson, Charles R. (Auteur), Saiago, Carlos M. (Auteur)
Format: Électronique eBook
Langue:Inglés
Publié: Cambridge, United Kingdom : Cambridge University Press, 2018.
Collection:Cambridge tracts in mathematics ; 211.
Sujets:
Eigenvalues.
Matrices.
Symmetric matrices.
Trees (Graph theory)
Valeurs propres.
Matrices symétriques.
Arbres (Théorie des graphes)
MATHEMATICS > Algebra > Intermediate.
Teoría de grafos
Matrices (Matemáticas)
Eigenvalues
Matrices
Symmetric matrices
Accès en ligne:Texto completo
Table des matières:
  • Background
  • Introduction
  • Parter-Wiener, etc. theory
  • Maximum multiplicity for trees, I
  • Multiple eigenvalues and structure
  • Maximum multiplicity, II
  • The minimum number of distinct eigenvalues
  • Construction techniques
  • Multiplicity lists for generalized stars
  • Double generalized stars
  • Linear trees
  • Nontrees
  • Geometric multiplicities for general matrices over a field.
  • Maximum Multiplicity for Trees, I
  • 3.1. Introduction
  • 3.2. Path Covers and Path Trees
  • 3.3. [delta](T) = Maximum p-q
  • 3.4. M(T) = P(T), [delta](T), n
  • mr(T)
  • 3.5. Calculation of M(T) and Bounds
  • 3.5.1. Calculation of M(T) in Linear Time
  • 3.5.2. Estimation of M(T) from the Degree Sequence of T
  • 4. Multiple Eigenvalues and Structure
  • 4.1. Perturbation of Diagonal Entries and Vertex Status
  • 4.2. Parter Vertices, Parter Sets and Fragmentation
  • 4.3. Fundamental Decomposition
  • 4.4. Eigenspace Structure and Vertex Classification
  • 4.5. Removal of an Edge
  • 4.5.1. Basic Inequalities
  • 4.5.2. Classification of Edges in Trees Based on the Classification of Their Vertices
  • 5. Maximum Multiplicity, II
  • 5.1. Structure of Matrices with a Maximum Multiplicity Eigenvalue
  • 5.2. NIM Trees
  • 5.3. Second Maximum Multiplicity
  • 6. Minimum Number of Distinct Eigenvalues
  • 6.1. Introduction
  • 6.2. Diameter and a Lower Bound for c(T)
  • 6.3. Method of Branch Duplication: Combinatorial and Algebraic
  • 6.4. Converse to the Diameter Lower Bound for Trees
  • 6.5. Trees of Diameter 7
  • 6.6. Function C(d) and Disparity
  • 6.7. Minimum Number of Multiplicities Equal to 1
  • 6.8. Relative Position of Multiple Eigenvalues in Ordered Lists
  • 6.8.1. Lower Bound for the Cardinality of a Fragmenting Parter Set
  • 6.8.2. Relative Position of a Single Multiple Eigenvalue
  • 6.8.3. Vertex Degrees
  • 6.8.4. Two Multiple Eigenvalues
  • 7. Construction Techniques
  • 7.1. Introduction
  • 7.2. Eigenvalues for Paths and Subpaths
  • 7.3. Method of Assignments
  • 7.4. Derivation of a Multiplicity List via Assignment: An Example
  • 7.5. 13-Vertex Example
  • 7.6. Implicit Function Theorem (IFT) Approach
  • 7.7. More IFT, Examples, Vines
  • 7.8. Polynomial Constructions
  • 8. Multiplicity Lists for Generalized Stars
  • 8.1. Introduction
  • 8.2. Characterization of Generalized Stars
  • 8.3. Case of Simple Stars
  • 8.4. Inverse Eigenvalue Problem for Generalized Stars
  • 8.5. Multiplicity Lists
  • 8.6. IEP versus Ordered Multiplicity Lists
  • 8.7. Upward Multiplicity Lists
  • 8.8. c(T) and U(T)
  • 9. Double Generalized Stars
  • 9.1. Introduction
  • 9.2. Observations about Double Generalized Stars
  • 9.3. Multiplicity Lists
  • 9.4. Double Paths
  • 10. Linear Trees
  • 10.1. Introduction
  • 10.2. Second Superposition Principle for Linear Trees
  • 10.3. Possible Multiplicity Lists for Linear Trees
  • 10.4. Cases of Sufficiency of Linear Trees
  • 10.5. Special Results for Linear Trees
  • 11. Nontrees
  • 11.1. Introduction and Observations
  • 11.2. Complete Graph
  • 11.3. Cycle
  • 11.4. Tree + an Edge
  • 11.4.1. Graph + an Edge
  • 11.5. Graphs G for Which M(G) = 2
  • 11.6. Graphs Permitting Just Two Distinct Eigenvalues
  • 11.7. Nearly Complete Graphs
  • 12. Geometric Multiplicities for General Matrices over a Field
  • 12.1. Preliminaries
  • 12.2. Geometric Parter-Wiener, etc. Theory
  • 12.3. Geometric Downer Branch Mechanism for General Matrices over a Field
  • 12.4. Maximum Geometric Multiplicity for a Tree
  • 12.5. Minimum Number of Distinct Eigenvalues in a Diagonalizable Matrix Whose Graph Is a Tree
  • Appendix A Multiplicity Lists for Trees on Fewer Than 12 Vertices
  • A.1. Tree on 3 Vertices (1 tree)
  • A.2. Trees on 4 Vertices (2 trees)
  • A.3. Trees on 5 Vertices (3 trees)
  • A.4. Trees.

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