Graphs and matrices:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
London [u.a.]
Springer [u.a.]
2014
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Universitext
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XI, 193 S. graph. Darst. |
| ISBN: | 9781447165682 |
Internformat
MARC
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| 100 | 1 | |a Bapat, Ravi B. |e Verfasser |0 (DE-588)115635831 |4 aut | |
| 245 | 1 | 0 | |a Graphs and matrices |c Ravindra B. Bapat |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a London [u.a.] |b Springer [u.a.] |c 2014 | |
| 300 | |a XI, 193 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 650 | 0 | 7 | |a Graphentheorie |0 (DE-588)4113782-6 |2 gnd |9 rswk-swf |
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| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027606928&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
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Datensatz im Suchindex
| _version_ | 1804152668726231040 |
|---|---|
| adam_text | Contents
1
Preliminaries
....................................... 1
1.1
Matrices
...................................... 1
1.2
Eigenvalues of Symmetric Matrices
.................... 5
1.3
Generalized Inverses
.............................. 8
1.4
Graphs
....................................... 10
References and Further Reading
........................... 11
2
Incidence Matrix
.................................... 13
2.1
Rank
......................................... 14
2.2
Minors
....................................... 15
2.3
Path Matrix
.................................... 17
2.4
Integer Generalized Inverses
......................... 18
2.5
Moore-Penrose Inverse
............................ 20
2.6 0-1
Incidence Matrix
............................. 22
2.7
Matchings in Bipartite Graphs
....................... 23
References and Further Reading
........................... 25
3
Adjacency Matrix
.................................... 27
3.1
Eigenvalues of Some Graphs
........................ 28
3.2
Determinant
.................................... 30
3.3
Bounds
....................................... 33
3.4
Energy of a Graph
............................... 38
3.5
Antiadjacency Matrix of a Directed Graph
............... 40
3.6
Nonsingular Trees
................................ 42
References and Further Readings
.......................... 48
4
Laplacian Matrix
.................................... 49
4.1
Basic Properties
................................. 50
4.2
Computing Laplacian Eigenvalues
..................... 51
4.3
Matrix-Tree Theorem
............................. 52
к
Contents
4.4
Bounds for Laplacian Spectral Radius
.................. 54
4.5
Edge-Laplacian of a Tree
........................... 55
References and Further Reading
........................... 59
5
Cycles and Cuts
..................................... 61
5.1
Fundamental Cycles and Fundamental Cuts
.............. 61
5.2
Fundamental Matrices
............................. 63
5.3
Minors
....................................... 64
References and Further Reading
........................... 68
6
Regular Graphs
..................................... 69
6.1
Perron-Frobenius Theory
........................... 69
6.2
Adjacency Algebra of a Regular Graph
................. 74
6.3
Complement and Line Graph of a Regular Graph
.......... 75
6.4
Strongly Regular Graphs and Friendship Theorem
......... 78
6.5
Graphs with Maximum Energy
....................... 81
References and Further Reading
........................... 85
7
Line Graph of a Tree
................................. 87
7.1
Block Graphs
................................... 87
7.2
Signless Laplacian Matrix
.......................... 93
7.3
Nullity of the Line Graph of a Tree
................... 95
References and Further Reading
........................... 99
8
Algebraic Connectivity
................................ 101
8.1
Preliminary Results
............................... 101
8.2
Classification of Trees
............................. 103
8.3
Monotonicity
Properties of Fiedler Vector
............... 108
8.4
Bounds for Algebraic Connectivity
....................
HO
References and Further Reading
........................... 114
9
Distance Matrix of a Tree
.............................. 115
9.1
Distance Matrix of a Graph
......................... 117
9.2
Distance Matrix and Laplacian of a Tree
................ 120
9.3
Eigenvalues of the Distance Matrix of a Tree
............. 125
References and Further Reading
........................... 130
10
Resistance Distance
................................... 131
10.1
The Triangle Inequality
............................ 132
10.2
Network Flows
.................................. 134
10.3
A Random Walk on Graphs
......................... 137
10.4
Effective Resistance in Electrical Networks
.............. 138
10.5
Resistance Matrix
................................ 139
References and Further Reading
........................... 144
Contents xi
11 Laplacian
Eigenvalues of Threshold Graphs
................ 145
11.1
Majorization
.................................... 145
11.2
Threshold Graphs
................................ 149
11.3
Spectral Integral Variation
.......................... 151
References and Further Reading
........................... 155
12
Positive Definite Completion Problem
..................... 157
12.1
Nonsingular Completion
........................... 157
12.2
Chordal Graphs
................................. 158
12.3
Positive Definite Completion
........................ 160
References and Further Reading
........................... 164
13
Matrix Games Based on Graphs
......................... 165
13.1
Matrix Games
.................................. 165
13.2
Vertex Selection Games
............................ 167
13.3
Tournament Games
............................... 169
13.4
Incidence Matrix Games
........................... 171
References and Further Reading
........................... 177
Hints and Solutions to Selected Exercises
...................... 179
Bibliography
........................................... 187
Index
................................................ 191
Universitext
Ravindra
В.
Bapaî
Graphs and Matrices
SecondEdition
This new edition illustrates the power of linear algebra in the study of graphs. The
emphasis on matrix techniques is greater than in other texts on algebraic graph theory.
Important matrices associated with graphs (for example, incidence, adjacency and
Laplacian matrices) are treated in detail.
Presenting a useful overview of selected topics in algebraic graph theory, early chapters
of the text focus on regular graphs, algebraic connectivity, the distance matrix of a
tree, and its generalized version for arbitrary graphs, known as the resistance matrix.
Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive
definite completion problem and matrix games based on a graph.
Such an extensive coverage of the subject area provides a welcome prompt for further
exploration. The inclusion of exercises enables practical learning throughout the book.
In the new edition, a new chapter is added on the line graph of a tree, while some
results in Chapter
6
on Perron-Frobenius theory are reorganized.
Whilst this book will be invaluable to students and researchers in graph theory and
combinatorial matrix theory, it will also benefit readers in the sciences and engineering.
|
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| id | DE-604.BV042167419 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:14:22Z |
| institution | BVB |
| isbn | 9781447165682 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027606928 |
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| owner_facet | DE-11 DE-355 DE-BY-UBR DE-29T DE-188 |
| physical | XI, 193 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Springer [u.a.] |
| record_format | marc |
| series2 | Universitext |
| spelling | Bapat, Ravi B. Verfasser (DE-588)115635831 aut Graphs and matrices Ravindra B. Bapat 2. ed. London [u.a.] Springer [u.a.] 2014 XI, 193 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Graphentheorie (DE-588)4113782-6 gnd rswk-swf Graphentheorie (DE-588)4113782-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4471-6569-9 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027606928&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027606928&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Bapat, Ravi B. Graphs and matrices Graphentheorie (DE-588)4113782-6 gnd |
| subject_GND | (DE-588)4113782-6 |
| title | Graphs and matrices |
| title_auth | Graphs and matrices |
| title_exact_search | Graphs and matrices |
| title_full | Graphs and matrices Ravindra B. Bapat |
| title_fullStr | Graphs and matrices Ravindra B. Bapat |
| title_full_unstemmed | Graphs and matrices Ravindra B. Bapat |
| title_short | Graphs and matrices |
| title_sort | graphs and matrices |
| topic | Graphentheorie (DE-588)4113782-6 gnd |
| topic_facet | Graphentheorie |
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