Title Eigenvalues, multiplicities and graphs / Charles R. Johnson, College of William and Mary, Williamsburg, Virginia ; Carlos M. Saiago, Universidade Nova de Lisboa

Published Cambridge, United Kingdom : Cambridge University Press, 2018 ©2018

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Description 1 online resource (xxii, 291 pages) : illustrations Series Cambridge tracts in mathematics ; 211 Cambridge tracts in mathematics ; 211. Contents 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 Summary This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues Bibliography Includes bibliographical references (pages 281-286) and index Notes Print version record Subject Eigenvalues. Matrices. Symmetric matrices. Trees (Graph theory) MATHEMATICS -- Algebra -- Intermediate. Teoría de grafos Matrices (Matemáticas) Eigenvalues Matrices Symmetric matrices Trees (Graph theory) Form Electronic book Author Saiago, Carlos M., author ISBN 9781108548137 110854813X

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