Simplicial complexes of graphs:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2008
|
| Schriftenreihe: | Lecture Notes in Mathematics
1928 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Teilw. zugl.: Stockholm, Royal Inst. of Technology, Diss., 2005 |
| Beschreibung: | XIV, 378 Seiten graph. Darst. |
| ISBN: | 9783540758587 |
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| 100 | 1 | |a Jonsson, Jakob |d 1972- |e Verfasser |0 (DE-588)1071864661 |4 aut | |
| 245 | 1 | 0 | |a Simplicial complexes of graphs |c Jakob Jonsson |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a XIV, 378 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture Notes in Mathematics |v 1928 | |
| 500 | |a Teilw. zugl.: Stockholm, Royal Inst. of Technology, Diss., 2005 | ||
| 650 | 4 | |a Algebra homológica | |
| 650 | 4 | |a Arboles de decisión | |
| 650 | 4 | |a Teoría Morse | |
| 650 | 4 | |a Teoría de grafos | |
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| 655 | 7 | |0 (DE-588)4113937-9 |a Hochschulschrift |2 gnd-content | |
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| 689 | 0 | 1 | |a Simplizialer Komplex |0 (DE-588)4181492-7 |D s |
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| 830 | 0 | |a Lecture Notes in Mathematics |v 1928 |w (DE-604)BV000676446 |9 1928 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016228800 | ||
Datensatz im Suchindex
| _version_ | 1804137246305026048 |
|---|---|
| adam_text | Contents
Part I Introduction and Basic Concepts
1 Introduction and Overview 3
1.1 Motivation and Background 6
1.1.1 Quillen Complexes 6
1.1.2 Minimal Free Resolutions of Certain Semigroup Algebras 7
1.1.3 Lie Algebras 8
1.1.4 Disconnected fc hypergraphs and Subspace Arrangements 9
1.1.5 Cohomology of Spaces of Knots 10
1.1.6 Determinantal Ideals 11
1.1.7 Other Examples 12
1.1.8 Links to Graph Theory 13
1.1.9 Complexity Theory and Evasiveness 14
1.2 Overview 14
2 Abstract Graphs and Set Systems 19
2.1 Graphs, Hypergraphs, and Digraphs 19
2.1.1 Graphs 20
2.1.2 Paths, Components and Cycles 20
2.1.3 Bipartite Graphs 21
2.1.4 Digraphs 21
2.1.5 Directed Paths and Cycles 21
2.1.6 Hypergraphs 22
2.1.7 General Terminology 22
2.2 Posets and Lattices 23
2.3 Abstract Simplicial Complexes 23
2.3.1 Basic Definitions 24
2.3.2 Dimension 24
2.3.3 Collapses 24
2.3.4 Joins, Cones, Suspensions, and Wedges 25
2.3.5 Alexander Duals 25
X Contents
2.3.6 Links and Deletions 25
2.3.7 Lifted Complexes 25
2.3.8 Order Complexes and Face Posets 25
2.3.9 Graph, Digraph, and Hypergraph Complexes
and Properties 26
2.4 Matroids 26
2.4.1 Graphic Matroids 27
2.5 Integer Partitions 28
3 Simplicial Topology 29
3.1 Simplicial Homology 29
3.2 Relative Homology 31
3.3 Homotopy Theory 32
3.4 Contractible Complexes and Their Relatives 35
3.4.1 Acyclic and fc acyclic Complexes 35
3.4.2 Contractible and A; connected Complexes 36
3.4.3 Collapsible Complexes 37
3.4.4 Nonevasive Complexes 38
3.5 Quotient Complexes 38
3.6 Shellable Complexes and Their Relatives 40
3.6.1 Cohen Macaulay Complexes 40
3.6.2 Constructive Complexes 41
3.6.3 Shellable Complexes 41
3.6.4 Vertex Decomposable Complexes 42
3.6.5 Topological Properties and Relations Between
Different Classes 43
3.7 Balls and Spheres 46
3.8 Stanley Reisner Rings 47
Part II Tools
4 Discrete Morse Theory 51
4.1 Informal Discussion 51
4.2 Acyclic Matchings 53
4.3 Simplicial Morse Theory 55
4.4 Discrete Morse Theory on Complexes of Groups 59
4.4.1 Independent Sets in the Homology of a Complex 61
4.4.2 Simple Applications 64
5 Decision Trees 67
5.1 Basic Properties of Decision Trees 69
5.1.1 Element Decision Trees 69
5.1.2 Set Decision Trees 70
5.2 Hierarchy of Almost Nonevasive Complexes 72
Contents XI
5.2.1 Semi nonevasive and Semi collapsible Complexes 73
5.2.2 Relations Between Some Important Glasses
of Complexes 76
5.3 Some Useful Constructions 79
5.4 Further Properties of Almost Nonevasive Complexes 81
5.5 A Potential Generalization 86
6 Miscellaneous Results 87
6.1 Posets 87
6.2 Depth 88
6.3 Vertex Decomposability 92
6.4 Enumeration 93
Part III Overview of Graph Complexes
7 Graph Properties 99
7.1 List of Complexes 100
7.2 Illustrations 104
8 Dihedral Graph Properties 107
8.1 Basic Definitions 108
8.2 List of Complexes 109
8.3 Illustrations Ill
9 Digraph Properties 113
9.1 List of Complexes 113
9.2 Illustrations 117
10 Main Goals and Proof Techniques 119
10.1 Homology 119
10.2 Homotopy Type 120
10.3 Connectivity Degree 120
10.4 Depth 120
10.5 Euler Characteristic 123
10.6 Remarks on Nonevasiveness and Related Properties 123
Part IV Vertex Degree
11 Matchings 127
11.1 Some General Results 128
11.2 Complete Graphs 130
11.2.1 Rational Homology 130
11.2.2 Homotopical Depth and Bottom Nonvanishing
Homology 131
XII Contents
11.2.3 Torsion in Higher Degree Homology Groups 136
11.2.4 Further Properties 141
11.3 Chessboards 143
11.3.1 Bottom Nonvanishing Homology 143
11.3.2 Torsion in Higher Degree Homology Groups 145
11.4 Paths and Cycles 148
12 Graphs of Bounded Degree 151
12.1 Bounded Degree Graphs Without Loops 152
12.1.1 The Case d = 2 153
12.1.2 The General Case 155
12.2 Bounded Degree Graphs with Loops 161
12.3 Euler Characteristic 165
Part V Cycles and Crossings
13 Forests and Matroids 171
13.1 Independence Complexes 172
13.2 Pseudo Independence Complexes 173
13.2.1 PI Graph Complexes 175
13.3 Strong Pseudo Independence Complexes 176
13.3.1 Sets in Matroids Avoiding Odd Cycles 181
13.3.2 SPI Graph Complexes 182
13.4 Alexander Duals of SPI Complexes 184
13.4.1 SPI* Monotone Graph Properties 186
14 Bipartite Graphs 189
14.1 Bipartite Graphs Without Restrictions 190
14.2 Disconnected Bipartite Graphs 192
14.3 Unbalanced Bipartite Graphs 193
14.3.1 Depth 194
14.3.2 Homotopy Type 195
14.3.3 Euler Characteristic 198
14.3.4 Generalization to Hypergraphs 203
15 Directed Variants of Forests and Bipartite Graphs 205
15.1 Directed Forests 206
15.2 Acyclic Digraphs 206
15.3 Bipartite Digraphs 208
15.4 Graded Digraphs 209
15.5 Digraphs with No Non alternating Circuits 213
15.6 Digraphs Without Odd Directed Cycles 213
Contents XIII
16 Noncrossing Graphs 217
16.1 The Associahedron 218
16.2 A Shelling of the Associahedron 219
16.3 Noncrossing Matchings 222
16.4 Noncrossing Forests 226
16.5 Noncrossing Bipartite Graphs 229
17 Non Hamiltonian Graphs 233
17.1 Homotopy Type 234
17.2 Homology 240
17.3 Directed Variant 242
Part VI Connectivity
18 Disconnected Graphs 245
18.1 Disconnected Graphs Without Restrictions 246
18.2 Graphs with No Large Components 247
18.2.1 Homotopy Type and Depth 248
18.2.2 Bottom Nonvanishing Homology Group 253
18.3 Graphs with Some Small Components 258
18.4 Graphs with Some Component of Size Not Divisible by p 262
18.5 Disconnected Hypergraphs 262
19 Not 2 connected Graphs 263
19.1 Homotopy Type 263
19.2 Homology 268
19.3 A Decision Tree 269
19.4 Generalization and Yet Another Proof 271
20 Not 3 connected Graphs and Beyond 275
20.1 Homotopy Type 275
20.2 Homology 283
20.3 A Related Polytopal Sphere 287
20.4 Not fc connected Graphs for k 3 289
21 Dihedral Variants of fe connected Graphs 291
21.1 A General Observation 292
21.2 Graphs with a Disconnected Polygon Representation 292
21.3 Graphs with a Separable Polygon Representation 294
21.4 Graphs with a Two separable Polygon Representation 298
22 Directed Variants of Connected Graphs 301
22.1 Not Strongly Connected Digraphs 301
22.2 Not Strongly 2 connected Digraphs 306
22.3 Non spanning Digraphs 307
XIV Contents
23 Not 2 edge connected Graphs 309
23.1 An Acyclic Matching 310
23.2 Enumerative Properties of the Given Matching 318
23.3 Bottom Nonvanishing Homology Group 321
23.4 Top Nonvanishing Homology Group 323
Part VII Cliques and Stable Sets
24 Graphs Avoiding fe matchings 329
25 t colorable Graphs 333
26 Graphs and Hypergraphs with Bounded
Covering Number 337
26.1 Solid Hypergraphs 338
26.2 A Related Simplicial Complex 340
26.3 An Acyclic Matching 341
26.4 Homotopy Type and Homology 343
26.5 Computations 347
26.6 Homotopical Depth 350
26.7 Triangle Free Graphs 351
26.8 Concluding Remarks and Open Problems 352
Part VIII Open Problems
27 Open Problems 357
References 363
Index 371
|
| adam_txt |
Contents
Part I Introduction and Basic Concepts
1 Introduction and Overview 3
1.1 Motivation and Background 6
1.1.1 Quillen Complexes 6
1.1.2 Minimal Free Resolutions of Certain Semigroup Algebras 7
1.1.3 Lie Algebras 8
1.1.4 Disconnected fc hypergraphs and Subspace Arrangements 9
1.1.5 Cohomology of Spaces of Knots 10
1.1.6 Determinantal Ideals 11
1.1.7 Other Examples 12
1.1.8 Links to Graph Theory 13
1.1.9 Complexity Theory and Evasiveness 14
1.2 Overview 14
2 Abstract Graphs and Set Systems 19
2.1 Graphs, Hypergraphs, and Digraphs 19
2.1.1 Graphs 20
2.1.2 Paths, Components and Cycles 20
2.1.3 Bipartite Graphs 21
2.1.4 Digraphs 21
2.1.5 Directed Paths and Cycles 21
2.1.6 Hypergraphs 22
2.1.7 General Terminology 22
2.2 Posets and Lattices 23
2.3 Abstract Simplicial Complexes 23
2.3.1 Basic Definitions 24
2.3.2 Dimension 24
2.3.3 Collapses 24
2.3.4 Joins, Cones, Suspensions, and Wedges 25
2.3.5 Alexander Duals 25
X Contents
2.3.6 Links and Deletions 25
2.3.7 Lifted Complexes 25
2.3.8 Order Complexes and Face Posets 25
2.3.9 Graph, Digraph, and Hypergraph Complexes
and Properties 26
2.4 Matroids 26
2.4.1 Graphic Matroids 27
2.5 Integer Partitions 28
3 Simplicial Topology 29
3.1 Simplicial Homology 29
3.2 Relative Homology 31
3.3 Homotopy Theory 32
3.4 Contractible Complexes and Their Relatives 35
3.4.1 Acyclic and fc acyclic Complexes 35
3.4.2 Contractible and A; connected Complexes 36
3.4.3 Collapsible Complexes 37
3.4.4 Nonevasive Complexes 38
3.5 Quotient Complexes 38
3.6 Shellable Complexes and Their Relatives 40
3.6.1 Cohen Macaulay Complexes 40
3.6.2 Constructive Complexes 41
3.6.3 Shellable Complexes 41
3.6.4 Vertex Decomposable Complexes 42
3.6.5 Topological Properties and Relations Between
Different Classes 43
3.7 Balls and Spheres 46
3.8 Stanley Reisner Rings 47
Part II Tools
4 Discrete Morse Theory 51
4.1 Informal Discussion 51
4.2 Acyclic Matchings 53
4.3 Simplicial Morse Theory 55
4.4 Discrete Morse Theory on Complexes of Groups 59
4.4.1 Independent Sets in the Homology of a Complex 61
4.4.2 Simple Applications 64
5 Decision Trees 67
5.1 Basic Properties of Decision Trees 69
5.1.1 Element Decision Trees 69
5.1.2 Set Decision Trees 70
5.2 Hierarchy of Almost Nonevasive Complexes 72
Contents XI
5.2.1 Semi nonevasive and Semi collapsible Complexes 73
5.2.2 Relations Between Some Important Glasses
of Complexes 76
5.3 Some Useful Constructions 79
5.4 Further Properties of Almost Nonevasive Complexes 81
5.5 A Potential Generalization 86
6 Miscellaneous Results 87
6.1 Posets 87
6.2 Depth 88
6.3 Vertex Decomposability 92
6.4 Enumeration 93
Part III Overview of Graph Complexes
7 Graph Properties 99
7.1 List of Complexes 100
7.2 Illustrations 104
8 Dihedral Graph Properties 107
8.1 Basic Definitions 108
8.2 List of Complexes 109
8.3 Illustrations Ill
9 Digraph Properties 113
9.1 List of Complexes 113
9.2 Illustrations 117
10 Main Goals and Proof Techniques 119
10.1 Homology 119
10.2 Homotopy Type 120
10.3 Connectivity Degree 120
10.4 Depth 120
10.5 Euler Characteristic 123
10.6 Remarks on Nonevasiveness and Related Properties 123
Part IV Vertex Degree
11 Matchings 127
11.1 Some General Results 128
11.2 Complete Graphs 130
11.2.1 Rational Homology 130
11.2.2 Homotopical Depth and Bottom Nonvanishing
Homology 131
XII Contents
11.2.3 Torsion in Higher Degree Homology Groups 136
11.2.4 Further Properties 141
11.3 Chessboards 143
11.3.1 Bottom Nonvanishing Homology 143
11.3.2 Torsion in Higher Degree Homology Groups 145
11.4 Paths and Cycles 148
12 Graphs of Bounded Degree 151
12.1 Bounded Degree Graphs Without Loops 152
12.1.1 The Case d = 2 153
12.1.2 The General Case 155
12.2 Bounded Degree Graphs with Loops 161
12.3 Euler Characteristic 165
Part V Cycles and Crossings
13 Forests and Matroids 171
13.1 Independence Complexes 172
13.2 Pseudo Independence Complexes 173
13.2.1 PI Graph Complexes 175
13.3 Strong Pseudo Independence Complexes 176
13.3.1 Sets in Matroids Avoiding Odd Cycles 181
13.3.2 SPI Graph Complexes 182
13.4 Alexander Duals of SPI Complexes 184
13.4.1 SPI* Monotone Graph Properties 186
14 Bipartite Graphs 189
14.1 Bipartite Graphs Without Restrictions 190
14.2 Disconnected Bipartite Graphs 192
14.3 Unbalanced Bipartite Graphs 193
14.3.1 Depth 194
14.3.2 Homotopy Type 195
14.3.3 Euler Characteristic 198
14.3.4 Generalization to Hypergraphs 203
15 Directed Variants of Forests and Bipartite Graphs 205
15.1 Directed Forests 206
15.2 Acyclic Digraphs 206
15.3 Bipartite Digraphs 208
15.4 Graded Digraphs 209
15.5 Digraphs with No Non alternating Circuits 213
15.6 Digraphs Without Odd Directed Cycles 213
Contents XIII
16 Noncrossing Graphs 217
16.1 The Associahedron 218
16.2 A Shelling of the Associahedron 219
16.3 Noncrossing Matchings 222
16.4 Noncrossing Forests 226
16.5 Noncrossing Bipartite Graphs 229
17 Non Hamiltonian Graphs 233
17.1 Homotopy Type 234
17.2 Homology 240
17.3 Directed Variant 242
Part VI Connectivity
18 Disconnected Graphs 245
18.1 Disconnected Graphs Without Restrictions 246
18.2 Graphs with No Large Components 247
18.2.1 Homotopy Type and Depth 248
18.2.2 Bottom Nonvanishing Homology Group 253
18.3 Graphs with Some Small Components 258
18.4 Graphs with Some Component of Size Not Divisible by p 262
18.5 Disconnected Hypergraphs 262
19 Not 2 connected Graphs 263
19.1 Homotopy Type 263
19.2 Homology 268
19.3 A Decision Tree 269
19.4 Generalization and Yet Another Proof 271
20 Not 3 connected Graphs and Beyond 275
20.1 Homotopy Type 275
20.2 Homology 283
20.3 A Related Polytopal Sphere 287
20.4 Not fc connected Graphs for k 3 289
21 Dihedral Variants of fe connected Graphs 291
21.1 A General Observation 292
21.2 Graphs with a Disconnected Polygon Representation 292
21.3 Graphs with a Separable Polygon Representation 294
21.4 Graphs with a Two separable Polygon Representation 298
22 Directed Variants of Connected Graphs 301
22.1 Not Strongly Connected Digraphs 301
22.2 Not Strongly 2 connected Digraphs 306
22.3 Non spanning Digraphs 307
XIV Contents
23 Not 2 edge connected Graphs 309
23.1 An Acyclic Matching 310
23.2 Enumerative Properties of the Given Matching 318
23.3 Bottom Nonvanishing Homology Group 321
23.4 Top Nonvanishing Homology Group 323
Part VII Cliques and Stable Sets
24 Graphs Avoiding fe matchings 329
25 t colorable Graphs 333
26 Graphs and Hypergraphs with Bounded
Covering Number 337
26.1 Solid Hypergraphs 338
26.2 A Related Simplicial Complex 340
26.3 An Acyclic Matching 341
26.4 Homotopy Type and Homology 343
26.5 Computations 347
26.6 Homotopical Depth 350
26.7 Triangle Free Graphs 351
26.8 Concluding Remarks and Open Problems 352
Part VIII Open Problems
27 Open Problems 357
References 363
Index 371 |
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| author | Jonsson, Jakob 1972- |
| author_GND | (DE-588)1071864661 |
| author_facet | Jonsson, Jakob 1972- |
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| author_sort | Jonsson, Jakob 1972- |
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| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 050f MAT 550f |
| ctrlnum | (OCoLC)427515206 (DE-599)BVBBV023024808 |
| dewey-full | 511/.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511/.5 |
| dewey-search | 511/.5 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| genre | (DE-588)4113937-9 Hochschulschrift gnd-content |
| genre_facet | Hochschulschrift |
| id | DE-604.BV023024808 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:14:38Z |
| indexdate | 2024-07-09T21:09:15Z |
| institution | BVB |
| isbn | 9783540758587 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016228800 |
| oclc_num | 427515206 |
| open_access_boolean | |
| owner | DE-824 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-83 DE-11 DE-188 DE-29T |
| owner_facet | DE-824 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-83 DE-11 DE-188 DE-29T |
| physical | XIV, 378 Seiten graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Lecture Notes in Mathematics |
| series2 | Lecture Notes in Mathematics |
| spelling | Jonsson, Jakob 1972- Verfasser (DE-588)1071864661 aut Simplicial complexes of graphs Jakob Jonsson Berlin [u.a.] Springer 2008 XIV, 378 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Mathematics 1928 Teilw. zugl.: Stockholm, Royal Inst. of Technology, Diss., 2005 Algebra homológica Arboles de decisión Teoría Morse Teoría de grafos Simplizialer Komplex (DE-588)4181492-7 gnd rswk-swf Graph (DE-588)4021842-9 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Graph (DE-588)4021842-9 s Simplizialer Komplex (DE-588)4181492-7 s DE-604 Lecture Notes in Mathematics 1928 (DE-604)BV000676446 1928 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016228800&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jonsson, Jakob 1972- Simplicial complexes of graphs Lecture Notes in Mathematics Algebra homológica Arboles de decisión Teoría Morse Teoría de grafos Simplizialer Komplex (DE-588)4181492-7 gnd Graph (DE-588)4021842-9 gnd |
| subject_GND | (DE-588)4181492-7 (DE-588)4021842-9 (DE-588)4113937-9 |
| title | Simplicial complexes of graphs |
| title_auth | Simplicial complexes of graphs |
| title_exact_search | Simplicial complexes of graphs |
| title_exact_search_txtP | Simplicial complexes of graphs |
| title_full | Simplicial complexes of graphs Jakob Jonsson |
| title_fullStr | Simplicial complexes of graphs Jakob Jonsson |
| title_full_unstemmed | Simplicial complexes of graphs Jakob Jonsson |
| title_short | Simplicial complexes of graphs |
| title_sort | simplicial complexes of graphs |
| topic | Algebra homológica Arboles de decisión Teoría Morse Teoría de grafos Simplizialer Komplex (DE-588)4181492-7 gnd Graph (DE-588)4021842-9 gnd |
| topic_facet | Algebra homológica Arboles de decisión Teoría Morse Teoría de grafos Simplizialer Komplex Graph Hochschulschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016228800&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT jonssonjakob simplicialcomplexesofgraphs |