Eigenvalues, multiplicities and graphs /:
This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues.
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom :
Cambridge University Press,
2018.
|
| Schriftenreihe: | Cambridge tracts in mathematics ;
211. |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues. |
| Beschreibung: | 1 online resource (xxii, 291 pages) : illustrations |
| Bibliographie: | Includes bibliographical references (pages 281-286) and index. |
| ISBN: | 9781108548137 110854813X |
Internformat
MARC
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| 082 | 7 | |a 512.9/434 |2 23 | |
| 049 | |a MAIN | ||
| 100 | 1 | |a Johnson, Charles R., |e author. |0 http://id.loc.gov/authorities/names/n85014264 | |
| 245 | 1 | 0 | |a Eigenvalues, multiplicities and graphs / |c Charles R. Johnson, College of William and Mary, Williamsburg, Virginia ; Carlos M. Saiago, Universidade Nova de Lisboa. |
| 264 | 1 | |a Cambridge, United Kingdom : |b Cambridge University Press, |c 2018. | |
| 264 | 4 | |c ©2018 | |
| 300 | |a 1 online resource (xxii, 291 pages) : |b illustrations | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Cambridge tracts in mathematics ; |v 211 | |
| 520 | |a This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues. | ||
| 504 | |a Includes bibliographical references (pages 281-286) and index. | ||
| 505 | 0 | 0 | |g Background -- |g Introduction -- |t Parter-Wiener, etc. theory -- |t Maximum multiplicity for trees, I -- |t Multiple eigenvalues and structure -- |t Maximum multiplicity, II -- |t The minimum number of distinct eigenvalues -- |t Construction techniques -- |t Multiplicity lists for generalized stars -- |t Double generalized stars -- Linear trees -- |t Nontrees -- |t Geometric multiplicities for general matrices over a field. |
| 505 | 0 | 0 | |g Maximum Multiplicity for Trees, I -- |g 3.1. |t Introduction -- |g 3.2. |t Path Covers and Path Trees -- |g 3.3. |t [delta](T) = Maximum p-q -- |g 3.4. |t M(T) = P(T), [delta](T), n -- mr(T) -- |g 3.5. |t Calculation of M(T) and Bounds -- |g 3.5.1. |t Calculation of M(T) in Linear Time -- |g 3.5.2. |t Estimation of M(T) from the Degree Sequence of T -- |g 4. |t Multiple Eigenvalues and Structure -- |g 4.1. |t Perturbation of Diagonal Entries and Vertex Status -- |g 4.2. |t Parter Vertices, Parter Sets and Fragmentation -- |g 4.3. |t Fundamental Decomposition -- |g 4.4. |t Eigenspace Structure and Vertex Classification -- |g 4.5. |t Removal of an Edge -- |g 4.5.1. |t Basic Inequalities -- |g 4.5.2. |t Classification of Edges in Trees Based on the Classification of Their Vertices -- |g 5. |t Maximum Multiplicity, II -- |g 5.1. |t Structure of Matrices with a Maximum Multiplicity Eigenvalue -- |g 5.2. |t NIM Trees -- |g 5.3. |t Second Maximum Multiplicity -- |g 6. |t Minimum Number of Distinct Eigenvalues -- |g 6.1. |t Introduction -- |g 6.2. |t Diameter and a Lower Bound for c(T) -- |g 6.3. |t Method of Branch Duplication: Combinatorial and Algebraic -- |g 6.4. |t Converse to the Diameter Lower Bound for Trees -- |g 6.5. |t Trees of Diameter 7 -- |g 6.6. |t Function C(d) and Disparity -- |g 6.7. |t Minimum Number of Multiplicities Equal to 1 -- |g 6.8. |t Relative Position of Multiple Eigenvalues in Ordered Lists -- |g 6.8.1. |t Lower Bound for the Cardinality of a Fragmenting Parter Set -- |g 6.8.2. |t Relative Position of a Single Multiple Eigenvalue -- |g 6.8.3. |t Vertex Degrees -- |g 6.8.4. |t Two Multiple Eigenvalues -- |g 7. |t Construction Techniques -- |g 7.1. |t Introduction -- |g 7.2. |t Eigenvalues for Paths and Subpaths -- |g 7.3. |t Method of Assignments -- |g 7.4. |t Derivation of a Multiplicity List via Assignment: An Example -- |g 7.5. |t 13-Vertex Example -- |g 7.6. |t Implicit Function Theorem (IFT) Approach -- |g 7.7. |t More IFT, Examples, Vines -- |g 7.8. |t Polynomial Constructions -- |g 8. |t Multiplicity Lists for Generalized Stars -- |g 8.1. |t Introduction -- |g 8.2. |t Characterization of Generalized Stars -- |g 8.3. |t Case of Simple Stars -- |g 8.4. |t Inverse Eigenvalue Problem for Generalized Stars -- |g 8.5. |t Multiplicity Lists -- |g 8.6. |t IEP versus Ordered Multiplicity Lists -- |g 8.7. |t Upward Multiplicity Lists -- |g 8.8. |t c(T) and U(T) -- |g 9. |t Double Generalized Stars -- |g 9.1. |t Introduction -- |g 9.2. |t Observations about Double Generalized Stars -- |g 9.3. |t Multiplicity Lists -- |g 9.4. |t Double Paths -- |g 10. |t Linear Trees -- |g 10.1. |t Introduction -- |g 10.2. |t Second Superposition Principle for Linear Trees -- |g 10.3. |t Possible Multiplicity Lists for Linear Trees -- |g 10.4. |t Cases of Sufficiency of Linear Trees -- |g 10.5. |t Special Results for Linear Trees -- |g 11. |t Nontrees -- |g 11.1. |t Introduction and Observations -- |g 11.2. |t Complete Graph -- |g 11.3. |t Cycle -- |g 11.4. |t Tree + an Edge -- |g 11.4.1. |t Graph + an Edge -- |g 11.5. |t Graphs G for Which M(G) = 2 -- |g 11.6. |t Graphs Permitting Just Two Distinct Eigenvalues -- |g 11.7. |t Nearly Complete Graphs -- |g 12. |t Geometric Multiplicities for General Matrices over a Field -- |g 12.1. |t Preliminaries -- |g 12.2. |t Geometric Parter-Wiener, etc. Theory -- |g 12.3. |t Geometric Downer Branch Mechanism for General Matrices over a Field -- |g 12.4. |t Maximum Geometric Multiplicity for a Tree -- |g 12.5. |t Minimum Number of Distinct Eigenvalues in a Diagonalizable Matrix Whose Graph Is a Tree -- |g Appendix |t A Multiplicity Lists for Trees on Fewer Than 12 Vertices -- |g A.1. |t Tree on 3 Vertices (1 tree) -- |g A.2. |t Trees on 4 Vertices (2 trees) -- |g A.3. |t Trees on 5 Vertices (3 trees) -- |g A.4. |t Trees. |
| 588 | 0 | |a Print version record. | |
| 650 | 0 | |a Eigenvalues. |0 http://id.loc.gov/authorities/subjects/sh85041389 | |
| 650 | 0 | |a Matrices. |0 http://id.loc.gov/authorities/subjects/sh85082210 | |
| 650 | 0 | |a Symmetric matrices. |0 http://id.loc.gov/authorities/subjects/sh85131435 | |
| 650 | 0 | |a Trees (Graph theory) |0 http://id.loc.gov/authorities/subjects/sh85137259 | |
| 650 | 6 | |a Valeurs propres. | |
| 650 | 6 | |a Matrices. | |
| 650 | 6 | |a Matrices symétriques. | |
| 650 | 6 | |a Arbres (Théorie des graphes) | |
| 650 | 7 | |a MATHEMATICS |x Algebra |x Intermediate. |2 bisacsh | |
| 650 | 7 | |a Teoría de grafos |2 embne | |
| 650 | 7 | |a Matrices (Matemáticas) |2 embne | |
| 650 | 7 | |a Eigenvalues |2 fast | |
| 650 | 7 | |a Matrices |2 fast | |
| 650 | 7 | |a Symmetric matrices |2 fast | |
| 650 | 7 | |a Trees (Graph theory) |2 fast | |
| 700 | 1 | |a Saiago, Carlos M., |e author. | |
| 776 | 0 | 8 | |i Print version: |a Johnson, Charles R. |t Eigenvalues, multiplicities and graphs. |d Cambridge, United Kingdom : Cambridge University Press, 2018 |z 110709545X |w (OCoLC)991790102 |
| 830 | 0 | |a Cambridge tracts in mathematics ; |v 211. |0 http://id.loc.gov/authorities/names/n42005726 | |
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Datensatz im Suchindex
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| _version_ | 1816882414312488961 |
| adam_text | |
| any_adam_object | |
| author | Johnson, Charles R. Saiago, Carlos M. |
| author_GND | http://id.loc.gov/authorities/names/n85014264 |
| author_facet | Johnson, Charles R. Saiago, Carlos M. |
| author_role | aut aut |
| author_sort | Johnson, Charles R. |
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| callnumber-raw | QA193 .J64 2018eb |
| callnumber-search | QA193 .J64 2018eb |
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| contents | 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. Introduction -- Path Covers and Path Trees -- [delta](T) = Maximum p-q -- M(T) = P(T), [delta](T), n -- mr(T) -- Calculation of M(T) and Bounds -- Calculation of M(T) in Linear Time -- Estimation of M(T) from the Degree Sequence of T -- Multiple Eigenvalues and Structure -- Perturbation of Diagonal Entries and Vertex Status -- Parter Vertices, Parter Sets and Fragmentation -- Fundamental Decomposition -- Eigenspace Structure and Vertex Classification -- Removal of an Edge -- Basic Inequalities -- Classification of Edges in Trees Based on the Classification of Their Vertices -- Maximum Multiplicity, II -- Structure of Matrices with a Maximum Multiplicity Eigenvalue -- NIM Trees -- Second Maximum Multiplicity -- Minimum Number of Distinct Eigenvalues -- Diameter and a Lower Bound for c(T) -- Method of Branch Duplication: Combinatorial and Algebraic -- Converse to the Diameter Lower Bound for Trees -- Trees of Diameter 7 -- Function C(d) and Disparity -- Minimum Number of Multiplicities Equal to 1 -- Relative Position of Multiple Eigenvalues in Ordered Lists -- Lower Bound for the Cardinality of a Fragmenting Parter Set -- Relative Position of a Single Multiple Eigenvalue -- Vertex Degrees -- Two Multiple Eigenvalues -- Construction Techniques -- Eigenvalues for Paths and Subpaths -- Method of Assignments -- Derivation of a Multiplicity List via Assignment: An Example -- 13-Vertex Example -- Implicit Function Theorem (IFT) Approach -- More IFT, Examples, Vines -- Polynomial Constructions -- Multiplicity Lists for Generalized Stars -- Characterization of Generalized Stars -- Case of Simple Stars -- Inverse Eigenvalue Problem for Generalized Stars -- Multiplicity Lists -- IEP versus Ordered Multiplicity Lists -- Upward Multiplicity Lists -- c(T) and U(T) -- Double Generalized Stars -- Observations about Double Generalized Stars -- Double Paths -- Linear Trees -- Second Superposition Principle for Linear Trees -- Possible Multiplicity Lists for Linear Trees -- Cases of Sufficiency of Linear Trees -- Special Results for Linear Trees -- Introduction and Observations -- Complete Graph -- Cycle -- Tree + an Edge -- Graph + an Edge -- Graphs G for Which M(G) = 2 -- Graphs Permitting Just Two Distinct Eigenvalues -- Nearly Complete Graphs -- Geometric Multiplicities for General Matrices over a Field -- Preliminaries -- Geometric Parter-Wiener, etc. Theory -- Geometric Downer Branch Mechanism for General Matrices over a Field -- Maximum Geometric Multiplicity for a Tree -- Minimum Number of Distinct Eigenvalues in a Diagonalizable Matrix Whose Graph Is a Tree -- A Multiplicity Lists for Trees on Fewer Than 12 Vertices -- Tree on 3 Vertices (1 tree) -- Trees on 4 Vertices (2 trees) -- Trees on 5 Vertices (3 trees) -- Trees. |
| ctrlnum | (OCoLC)1025334932 |
| dewey-full | 512.9/434 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.9/434 |
| dewey-search | 512.9/434 |
| dewey-sort | 3512.9 3434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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--</subfield><subfield code="g">5.1.</subfield><subfield code="t">Structure of Matrices with a Maximum Multiplicity Eigenvalue --</subfield><subfield code="g">5.2.</subfield><subfield code="t">NIM Trees --</subfield><subfield code="g">5.3.</subfield><subfield code="t">Second Maximum Multiplicity --</subfield><subfield code="g">6.</subfield><subfield code="t">Minimum Number of Distinct Eigenvalues --</subfield><subfield code="g">6.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">6.2.</subfield><subfield code="t">Diameter and a Lower Bound for c(T) --</subfield><subfield code="g">6.3.</subfield><subfield code="t">Method of Branch Duplication: Combinatorial and Algebraic --</subfield><subfield code="g">6.4.</subfield><subfield code="t">Converse to the Diameter Lower Bound for Trees --</subfield><subfield code="g">6.5.</subfield><subfield code="t">Trees of Diameter 7 --</subfield><subfield code="g">6.6.</subfield><subfield code="t">Function C(d) and Disparity --</subfield><subfield code="g">6.7.</subfield><subfield code="t">Minimum Number of Multiplicities Equal to 1 --</subfield><subfield code="g">6.8.</subfield><subfield code="t">Relative Position of Multiple Eigenvalues in Ordered Lists --</subfield><subfield code="g">6.8.1.</subfield><subfield code="t">Lower Bound for the Cardinality of a Fragmenting Parter Set --</subfield><subfield code="g">6.8.2.</subfield><subfield code="t">Relative Position of a Single Multiple Eigenvalue --</subfield><subfield code="g">6.8.3.</subfield><subfield code="t">Vertex Degrees --</subfield><subfield code="g">6.8.4.</subfield><subfield code="t">Two Multiple Eigenvalues --</subfield><subfield code="g">7.</subfield><subfield code="t">Construction Techniques --</subfield><subfield code="g">7.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">7.2.</subfield><subfield code="t">Eigenvalues for Paths and Subpaths --</subfield><subfield code="g">7.3.</subfield><subfield code="t">Method of Assignments --</subfield><subfield code="g">7.4.</subfield><subfield code="t">Derivation of a Multiplicity List via Assignment: An Example --</subfield><subfield code="g">7.5.</subfield><subfield code="t">13-Vertex Example --</subfield><subfield code="g">7.6.</subfield><subfield code="t">Implicit Function Theorem (IFT) Approach --</subfield><subfield code="g">7.7.</subfield><subfield code="t">More IFT, Examples, Vines --</subfield><subfield code="g">7.8.</subfield><subfield code="t">Polynomial Constructions --</subfield><subfield code="g">8.</subfield><subfield code="t">Multiplicity Lists for Generalized Stars --</subfield><subfield code="g">8.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">8.2.</subfield><subfield code="t">Characterization of Generalized Stars --</subfield><subfield code="g">8.3.</subfield><subfield code="t">Case of Simple Stars --</subfield><subfield code="g">8.4.</subfield><subfield code="t">Inverse Eigenvalue Problem for Generalized Stars --</subfield><subfield code="g">8.5.</subfield><subfield code="t">Multiplicity Lists --</subfield><subfield code="g">8.6.</subfield><subfield code="t">IEP versus Ordered Multiplicity Lists --</subfield><subfield code="g">8.7.</subfield><subfield code="t">Upward Multiplicity Lists --</subfield><subfield code="g">8.8.</subfield><subfield code="t">c(T) and U(T) --</subfield><subfield code="g">9.</subfield><subfield code="t">Double Generalized Stars --</subfield><subfield code="g">9.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">9.2.</subfield><subfield code="t">Observations about Double Generalized Stars --</subfield><subfield code="g">9.3.</subfield><subfield code="t">Multiplicity Lists --</subfield><subfield code="g">9.4.</subfield><subfield code="t">Double Paths --</subfield><subfield code="g">10.</subfield><subfield code="t">Linear Trees --</subfield><subfield code="g">10.1.</subfield><subfield code="t">Introduction --</subfield><subfield code="g">10.2.</subfield><subfield code="t">Second Superposition Principle for Linear Trees --</subfield><subfield code="g">10.3.</subfield><subfield code="t">Possible Multiplicity Lists for Linear Trees --</subfield><subfield code="g">10.4.</subfield><subfield code="t">Cases of Sufficiency of Linear Trees --</subfield><subfield code="g">10.5.</subfield><subfield code="t">Special Results for Linear Trees --</subfield><subfield code="g">11.</subfield><subfield code="t">Nontrees --</subfield><subfield code="g">11.1.</subfield><subfield code="t">Introduction and Observations --</subfield><subfield code="g">11.2.</subfield><subfield code="t">Complete Graph --</subfield><subfield code="g">11.3.</subfield><subfield code="t">Cycle --</subfield><subfield code="g">11.4.</subfield><subfield code="t">Tree + an Edge --</subfield><subfield code="g">11.4.1.</subfield><subfield code="t">Graph + an Edge --</subfield><subfield code="g">11.5.</subfield><subfield code="t">Graphs G for Which M(G) = 2 --</subfield><subfield code="g">11.6.</subfield><subfield code="t">Graphs Permitting Just Two Distinct Eigenvalues --</subfield><subfield code="g">11.7.</subfield><subfield code="t">Nearly Complete Graphs --</subfield><subfield code="g">12.</subfield><subfield code="t">Geometric Multiplicities for General Matrices over a Field --</subfield><subfield code="g">12.1.</subfield><subfield code="t">Preliminaries --</subfield><subfield code="g">12.2.</subfield><subfield code="t">Geometric Parter-Wiener, etc. Theory --</subfield><subfield code="g">12.3.</subfield><subfield code="t">Geometric Downer Branch Mechanism for General Matrices over a Field --</subfield><subfield code="g">12.4.</subfield><subfield code="t">Maximum Geometric Multiplicity for a Tree --</subfield><subfield code="g">12.5.</subfield><subfield code="t">Minimum Number of Distinct Eigenvalues in a Diagonalizable Matrix Whose Graph Is a Tree --</subfield><subfield code="g">Appendix</subfield><subfield code="t">A Multiplicity 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| id | ZDB-4-EBA-on1025334932 |
| illustrated | Illustrated |
| indexdate | 2024-11-27T13:28:14Z |
| institution | BVB |
| isbn | 9781108548137 110854813X |
| language | English |
| oclc_num | 1025334932 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (xxii, 291 pages) : illustrations |
| psigel | ZDB-4-EBA |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | Cambridge University Press, |
| record_format | marc |
| series | Cambridge tracts in mathematics ; |
| series2 | Cambridge tracts in mathematics ; |
| spelling | Johnson, Charles R., author. http://id.loc.gov/authorities/names/n85014264 Eigenvalues, multiplicities and graphs / Charles R. Johnson, College of William and Mary, Williamsburg, Virginia ; Carlos M. Saiago, Universidade Nova de Lisboa. Cambridge, United Kingdom : Cambridge University Press, 2018. ©2018 1 online resource (xxii, 291 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier Cambridge tracts in mathematics ; 211 This book investigates the influence of the graph of a symmetric matrix on the multiplicities of its eigenvalues. Includes bibliographical references (pages 281-286) and index. 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. Print version record. Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Matrices. http://id.loc.gov/authorities/subjects/sh85082210 Symmetric matrices. http://id.loc.gov/authorities/subjects/sh85131435 Trees (Graph theory) http://id.loc.gov/authorities/subjects/sh85137259 Valeurs propres. Matrices. Matrices symétriques. Arbres (Théorie des graphes) MATHEMATICS Algebra Intermediate. bisacsh Teoría de grafos embne Matrices (Matemáticas) embne Eigenvalues fast Matrices fast Symmetric matrices fast Trees (Graph theory) fast Saiago, Carlos M., author. Print version: Johnson, Charles R. Eigenvalues, multiplicities and graphs. Cambridge, United Kingdom : Cambridge University Press, 2018 110709545X (OCoLC)991790102 Cambridge tracts in mathematics ; 211. http://id.loc.gov/authorities/names/n42005726 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=1694366 Volltext |
| spellingShingle | Johnson, Charles R. Saiago, Carlos M. Eigenvalues, multiplicities and graphs / Cambridge tracts in mathematics ; 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. Introduction -- Path Covers and Path Trees -- [delta](T) = Maximum p-q -- M(T) = P(T), [delta](T), n -- mr(T) -- Calculation of M(T) and Bounds -- Calculation of M(T) in Linear Time -- Estimation of M(T) from the Degree Sequence of T -- Multiple Eigenvalues and Structure -- Perturbation of Diagonal Entries and Vertex Status -- Parter Vertices, Parter Sets and Fragmentation -- Fundamental Decomposition -- Eigenspace Structure and Vertex Classification -- Removal of an Edge -- Basic Inequalities -- Classification of Edges in Trees Based on the Classification of Their Vertices -- Maximum Multiplicity, II -- Structure of Matrices with a Maximum Multiplicity Eigenvalue -- NIM Trees -- Second Maximum Multiplicity -- Minimum Number of Distinct Eigenvalues -- Diameter and a Lower Bound for c(T) -- Method of Branch Duplication: Combinatorial and Algebraic -- Converse to the Diameter Lower Bound for Trees -- Trees of Diameter 7 -- Function C(d) and Disparity -- Minimum Number of Multiplicities Equal to 1 -- Relative Position of Multiple Eigenvalues in Ordered Lists -- Lower Bound for the Cardinality of a Fragmenting Parter Set -- Relative Position of a Single Multiple Eigenvalue -- Vertex Degrees -- Two Multiple Eigenvalues -- Construction Techniques -- Eigenvalues for Paths and Subpaths -- Method of Assignments -- Derivation of a Multiplicity List via Assignment: An Example -- 13-Vertex Example -- Implicit Function Theorem (IFT) Approach -- More IFT, Examples, Vines -- Polynomial Constructions -- Multiplicity Lists for Generalized Stars -- Characterization of Generalized Stars -- Case of Simple Stars -- Inverse Eigenvalue Problem for Generalized Stars -- Multiplicity Lists -- IEP versus Ordered Multiplicity Lists -- Upward Multiplicity Lists -- c(T) and U(T) -- Double Generalized Stars -- Observations about Double Generalized Stars -- Double Paths -- Linear Trees -- Second Superposition Principle for Linear Trees -- Possible Multiplicity Lists for Linear Trees -- Cases of Sufficiency of Linear Trees -- Special Results for Linear Trees -- Introduction and Observations -- Complete Graph -- Cycle -- Tree + an Edge -- Graph + an Edge -- Graphs G for Which M(G) = 2 -- Graphs Permitting Just Two Distinct Eigenvalues -- Nearly Complete Graphs -- Geometric Multiplicities for General Matrices over a Field -- Preliminaries -- Geometric Parter-Wiener, etc. Theory -- Geometric Downer Branch Mechanism for General Matrices over a Field -- Maximum Geometric Multiplicity for a Tree -- Minimum Number of Distinct Eigenvalues in a Diagonalizable Matrix Whose Graph Is a Tree -- A Multiplicity Lists for Trees on Fewer Than 12 Vertices -- Tree on 3 Vertices (1 tree) -- Trees on 4 Vertices (2 trees) -- Trees on 5 Vertices (3 trees) -- Trees. Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Matrices. http://id.loc.gov/authorities/subjects/sh85082210 Symmetric matrices. http://id.loc.gov/authorities/subjects/sh85131435 Trees (Graph theory) http://id.loc.gov/authorities/subjects/sh85137259 Valeurs propres. Matrices. Matrices symétriques. Arbres (Théorie des graphes) MATHEMATICS Algebra Intermediate. bisacsh Teoría de grafos embne Matrices (Matemáticas) embne Eigenvalues fast Matrices fast Symmetric matrices fast Trees (Graph theory) fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85041389 http://id.loc.gov/authorities/subjects/sh85082210 http://id.loc.gov/authorities/subjects/sh85131435 http://id.loc.gov/authorities/subjects/sh85137259 |
| title | Eigenvalues, multiplicities and graphs / |
| title_alt | 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. Introduction -- Path Covers and Path Trees -- [delta](T) = Maximum p-q -- M(T) = P(T), [delta](T), n -- mr(T) -- Calculation of M(T) and Bounds -- Calculation of M(T) in Linear Time -- Estimation of M(T) from the Degree Sequence of T -- Multiple Eigenvalues and Structure -- Perturbation of Diagonal Entries and Vertex Status -- Parter Vertices, Parter Sets and Fragmentation -- Fundamental Decomposition -- Eigenspace Structure and Vertex Classification -- Removal of an Edge -- Basic Inequalities -- Classification of Edges in Trees Based on the Classification of Their Vertices -- Maximum Multiplicity, II -- Structure of Matrices with a Maximum Multiplicity Eigenvalue -- NIM Trees -- Second Maximum Multiplicity -- Minimum Number of Distinct Eigenvalues -- Diameter and a Lower Bound for c(T) -- Method of Branch Duplication: Combinatorial and Algebraic -- Converse to the Diameter Lower Bound for Trees -- Trees of Diameter 7 -- Function C(d) and Disparity -- Minimum Number of Multiplicities Equal to 1 -- Relative Position of Multiple Eigenvalues in Ordered Lists -- Lower Bound for the Cardinality of a Fragmenting Parter Set -- Relative Position of a Single Multiple Eigenvalue -- Vertex Degrees -- Two Multiple Eigenvalues -- Construction Techniques -- Eigenvalues for Paths and Subpaths -- Method of Assignments -- Derivation of a Multiplicity List via Assignment: An Example -- 13-Vertex Example -- Implicit Function Theorem (IFT) Approach -- More IFT, Examples, Vines -- Polynomial Constructions -- Multiplicity Lists for Generalized Stars -- Characterization of Generalized Stars -- Case of Simple Stars -- Inverse Eigenvalue Problem for Generalized Stars -- Multiplicity Lists -- IEP versus Ordered Multiplicity Lists -- Upward Multiplicity Lists -- c(T) and U(T) -- Double Generalized Stars -- Observations about Double Generalized Stars -- Double Paths -- Linear Trees -- Second Superposition Principle for Linear Trees -- Possible Multiplicity Lists for Linear Trees -- Cases of Sufficiency of Linear Trees -- Special Results for Linear Trees -- Introduction and Observations -- Complete Graph -- Cycle -- Tree + an Edge -- Graph + an Edge -- Graphs G for Which M(G) = 2 -- Graphs Permitting Just Two Distinct Eigenvalues -- Nearly Complete Graphs -- Geometric Multiplicities for General Matrices over a Field -- Preliminaries -- Geometric Parter-Wiener, etc. Theory -- Geometric Downer Branch Mechanism for General Matrices over a Field -- Maximum Geometric Multiplicity for a Tree -- Minimum Number of Distinct Eigenvalues in a Diagonalizable Matrix Whose Graph Is a Tree -- A Multiplicity Lists for Trees on Fewer Than 12 Vertices -- Tree on 3 Vertices (1 tree) -- Trees on 4 Vertices (2 trees) -- Trees on 5 Vertices (3 trees) -- Trees. |
| title_auth | Eigenvalues, multiplicities and graphs / |
| title_exact_search | Eigenvalues, multiplicities and graphs / |
| title_full | Eigenvalues, multiplicities and graphs / Charles R. Johnson, College of William and Mary, Williamsburg, Virginia ; Carlos M. Saiago, Universidade Nova de Lisboa. |
| title_fullStr | Eigenvalues, multiplicities and graphs / Charles R. Johnson, College of William and Mary, Williamsburg, Virginia ; Carlos M. Saiago, Universidade Nova de Lisboa. |
| title_full_unstemmed | Eigenvalues, multiplicities and graphs / Charles R. Johnson, College of William and Mary, Williamsburg, Virginia ; Carlos M. Saiago, Universidade Nova de Lisboa. |
| title_short | Eigenvalues, multiplicities and graphs / |
| title_sort | eigenvalues multiplicities and graphs |
| topic | Eigenvalues. http://id.loc.gov/authorities/subjects/sh85041389 Matrices. http://id.loc.gov/authorities/subjects/sh85082210 Symmetric matrices. http://id.loc.gov/authorities/subjects/sh85131435 Trees (Graph theory) http://id.loc.gov/authorities/subjects/sh85137259 Valeurs propres. Matrices. Matrices symétriques. Arbres (Théorie des graphes) MATHEMATICS Algebra Intermediate. bisacsh Teoría de grafos embne Matrices (Matemáticas) embne Eigenvalues fast Matrices fast Symmetric matrices fast Trees (Graph theory) fast |
| topic_facet | 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 |
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