Eigenvalues, multiplicities and graphs /
This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues.
Cote: | Libro Electrónico |
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Auteurs principaux: | , |
Format: | Électronique eBook |
Langue: | Inglés |
Publié: |
Cambridge, United Kingdom :
Cambridge University Press,
2018.
|
Collection: | Cambridge tracts in mathematics ;
211. |
Sujets: | |
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.