Laplacian eigenvectors of graphs: Perron-Frobenius and Faber-Krahn type theorems
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2007
|
| Series: | Lecture notes in mathematics
1915 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 115 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783540735090 3540735097 |
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MARC
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| 100 | 1 | |a Bıyıkoğlu, Türker |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Laplacian eigenvectors of graphs |b Perron-Frobenius and Faber-Krahn type theorems |c Türker Bıyıkoğlu ; Josef Leydold ; Peter F. Stadler |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a VIII, 115 S. |b graph. Darst. |c 235 mm x 155 mm | ||
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| 650 | 7 | |a Operadores |2 larpcal | |
| 650 | 7 | |a Teoria dos grafos |2 larpcal | |
| 650 | 4 | |a Vecteurs | |
| 650 | 4 | |a Eigenvectors | |
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Record in the Search Index
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| adam_text | Contents
Introduction
............................................... 1
1.1 Matrix
Representations of a Graph
........................ 2
1.2
Finite Differences
........................................ 4
1.3
Landscapes on Graphs
................................... 4
1.4
Related Matrices
........................................ 7
1.5
Graphs with a Boundary: The Discrete Dirichlet Problem
.... 8
1.6
Generalized Graph Laplacians
............................. 10
1.7
Colin
de Verdiére
Matrices
................................ 11
1.8
Practical Applications of Laplacians Eigenvectors
............ 12
Graph Laplacians
.......................................... 15
2.1
Basic Properties of Graph Laplacians
...................... 15
2.2
Weighted Graphs
........................................ 17
2.3
The Rayleigh Quotient
................................... 18
2.4
Calculus on Graphs
...................................... 19
2.5
Basic Properties of Eigenfunctions
......................... 20
2.6
Graph Automorphisms and Eigenfunctions
................. 22
2.7
Quasi-Abelian Cayley Graphs
............................. 23
2.8
The Perron-Frobenius Theorem
........................... 26
Eigenfunctions and Nodal Domains
........................ 29
3.1
Couranťs
Nodal Domain Theorem
......................... 29
3.2
Proof of the Nodal Domain Theorem
....................... 33
3.3
Algebraic Connectivity. Fiedler Vectors and Perron Branches
.. 35
3.4
Some Results for Multiple Eigenvalues
..................... 39
3.5
The Courant-Herrmann Conjecture
........................ 41
3.6
Improvements for Special Cases
........................... 42
3.7
Faria
Vectors and Minimum Number of Nodal Domains
...... 46
3.8
Sign Pattern and Nodal Domains
.......................... 47
VIII Contents
4
Nodal
Domain Theorems
for
Special Graph
Classes........
49
4.1
Trees
.................................................. 49
4.2 Cographs
and Threshold Graphs
.......................... 54
4.3
Product Graphs and the Boolean Hypercube
................ 58
5
Computational Experiments
............................... 67
5.1
Nodal Domains and
Hyperplane
Arrangements
.............. 67
5.2
A Hillclimbing Algorithm
................................. 69
5.3
Numerical Experiments for the Boolean Hypercube
.......... 70
5.4
Local Optima
........................................... 73
6
Faber-Krahn Type Inequalities
............................ 77
6.1
Basic Properties of Dirichlet Operators
..................... 78
6.2
The Faber-Krahn Property
............................... 79
6.3
Unweighted Trees
....................................... 81
6.4
Semiregular Trees
....................................... 84
6.5
Rearrangements and Dirichlet Operators
................... 86
6.6
Perturbations and Branches
............................... 90
A Basic Notations
............................................ 93
В
Eigenfunctions Used in Figures
............................ 97
С
List of Symbols
............................................ 99
References
.....................................................101
Index
..........................................................113
|
| adam_txt |
Contents
Introduction
. 1
1.1 Matrix
Representations of a Graph
. 2
1.2
Finite Differences
. 4
1.3
Landscapes on Graphs
. 4
1.4
Related Matrices
. 7
1.5
Graphs with a Boundary: The Discrete Dirichlet Problem
. 8
1.6
Generalized Graph Laplacians
. 10
1.7
Colin
de Verdiére
Matrices
. 11
1.8
Practical Applications of Laplacians Eigenvectors
. 12
Graph Laplacians
. 15
2.1
Basic Properties of Graph Laplacians
. 15
2.2
Weighted Graphs
. 17
2.3
The Rayleigh Quotient
. 18
2.4
Calculus on Graphs
. 19
2.5
Basic Properties of Eigenfunctions
. 20
2.6
Graph Automorphisms and Eigenfunctions
. 22
2.7
Quasi-Abelian Cayley Graphs
. 23
2.8
The Perron-Frobenius Theorem
. 26
Eigenfunctions and Nodal Domains
. 29
3.1
Couranťs
Nodal Domain Theorem
. 29
3.2
Proof of the Nodal Domain Theorem
. 33
3.3
Algebraic Connectivity. Fiedler Vectors and Perron Branches
. 35
3.4
Some Results for Multiple Eigenvalues
. 39
3.5
The Courant-Herrmann Conjecture
. 41
3.6
Improvements for Special Cases
. 42
3.7
Faria
Vectors and Minimum Number of Nodal Domains
. 46
3.8
Sign Pattern and Nodal Domains
. 47
VIII Contents
4
Nodal
Domain Theorems
for
Special Graph
Classes.
49
4.1
Trees
. 49
4.2 Cographs
and Threshold Graphs
. 54
4.3
Product Graphs and the Boolean Hypercube
. 58
5
Computational Experiments
. 67
5.1
Nodal Domains and
Hyperplane
Arrangements
. 67
5.2
A Hillclimbing Algorithm
. 69
5.3
Numerical Experiments for the Boolean Hypercube
. 70
5.4
Local Optima
. 73
6
Faber-Krahn Type Inequalities
. 77
6.1
Basic Properties of Dirichlet Operators
. 78
6.2
The Faber-Krahn Property
. 79
6.3
Unweighted Trees
. 81
6.4
Semiregular Trees
. 84
6.5
Rearrangements and Dirichlet Operators
. 86
6.6
Perturbations and Branches
. 90
A Basic Notations
. 93
В
Eigenfunctions Used in Figures
. 97
С
List of Symbols
. 99
References
.101
Index
.113 |
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| author | Bıyıkoğlu, Türker Leydold, Josef Stadler, Peter F. 1975- |
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| ctrlnum | (OCoLC)164393719 (DE-599)DNB984632506 |
| dewey-full | 512.94/34 512.9/436 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| id | DE-604.BV022619489 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:19:36Z |
| indexdate | 2024-07-09T21:01:49Z |
| institution | BVB |
| isbn | 9783540735090 3540735097 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015825575 |
| oclc_num | 164393719 |
| open_access_boolean | |
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| owner_facet | DE-824 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-83 DE-188 |
| physical | VIII, 115 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Bıyıkoğlu, Türker Verfasser aut Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems Türker Bıyıkoğlu ; Josef Leydold ; Peter F. Stadler Berlin [u.a.] Springer 2007 VIII, 115 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1915 Grafentheorie gtt Graphes, Théorie des Laplace-operatoren gtt Laplacien Operadores larpcal Teoria dos grafos larpcal Vecteurs Eigenvectors Graph theory Laplacian operator Graph (DE-588)4021842-9 gnd rswk-swf Laplace-Operator (DE-588)4166772-4 gnd rswk-swf Eigenvektor (DE-588)4151198-0 gnd rswk-swf Graph (DE-588)4021842-9 s Laplace-Operator (DE-588)4166772-4 s Eigenvektor (DE-588)4151198-0 s DE-604 Leydold, Josef Verfasser aut Stadler, Peter F. 1975- Verfasser (DE-588)112831222 aut Lecture notes in mathematics 1915 (DE-604)BV000676446 1915 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015825575&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bıyıkoğlu, Türker Leydold, Josef Stadler, Peter F. 1975- Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems Lecture notes in mathematics Grafentheorie gtt Graphes, Théorie des Laplace-operatoren gtt Laplacien Operadores larpcal Teoria dos grafos larpcal Vecteurs Eigenvectors Graph theory Laplacian operator Graph (DE-588)4021842-9 gnd Laplace-Operator (DE-588)4166772-4 gnd Eigenvektor (DE-588)4151198-0 gnd |
| subject_GND | (DE-588)4021842-9 (DE-588)4166772-4 (DE-588)4151198-0 |
| title | Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems |
| title_auth | Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems |
| title_exact_search | Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems |
| title_exact_search_txtP | Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems |
| title_full | Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems Türker Bıyıkoğlu ; Josef Leydold ; Peter F. Stadler |
| title_fullStr | Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems Türker Bıyıkoğlu ; Josef Leydold ; Peter F. Stadler |
| title_full_unstemmed | Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems Türker Bıyıkoğlu ; Josef Leydold ; Peter F. Stadler |
| title_short | Laplacian eigenvectors of graphs |
| title_sort | laplacian eigenvectors of graphs perron frobenius and faber krahn type theorems |
| title_sub | Perron-Frobenius and Faber-Krahn type theorems |
| topic | Grafentheorie gtt Graphes, Théorie des Laplace-operatoren gtt Laplacien Operadores larpcal Teoria dos grafos larpcal Vecteurs Eigenvectors Graph theory Laplacian operator Graph (DE-588)4021842-9 gnd Laplace-Operator (DE-588)4166772-4 gnd Eigenvektor (DE-588)4151198-0 gnd |
| topic_facet | Grafentheorie Graphes, Théorie des Laplace-operatoren Laplacien Operadores Teoria dos grafos Vecteurs Eigenvectors Graph theory Laplacian operator Graph Laplace-Operator Eigenvektor |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015825575&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT bıyıkogluturker laplacianeigenvectorsofgraphsperronfrobeniusandfaberkrahntypetheorems AT leydoldjosef laplacianeigenvectorsofgraphsperronfrobeniusandfaberkrahntypetheorems AT stadlerpeterf laplacianeigenvectorsofgraphsperronfrobeniusandfaberkrahntypetheorems |