Algebraic graph theory: morphisms, monoids and matrices
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2011
|
| Series: | De Gruyter studies in mathematics
41 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVI, 308 S. graph. Darst. 240 mm x 170 mm |
| ISBN: | 3110254085 9783110254082 |
Staff View
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| 245 | 1 | 0 | |a Algebraic graph theory |b morphisms, monoids and matrices |c Ulrich Knauer |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2011 | |
| 300 | |a XVI, 308 S. |b graph. Darst. |c 240 mm x 170 mm | ||
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Record in the Search Index
| _version_ | 1805145057862352896 |
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| adam_text |
IMAGE 1
CONTENTS
PREFACE V
1 DIRECTED AND UNDIRECTED GRAPHS 1
1 . 1 FORMAL DESCRIPTION OF GRAPHS 1
1.2 CONNECTEDNESS AND EQUIVALENCE RELATIONS 4
1.3 SOME SPECIAL GRAPHS 5
1.4 HOMOMORPHISMS 7
1.5 HALF-, LOCALLY, QUASI-STRONG AND METRIC HOMOMORPHISMS 11 1.6 THE
FACTOR GRAPH, CONGRUENCES, AND THE HOMOMORPHISM THEOREM . . 14 FACTOR
GRAPHS 14
THE HOMOMORPHISM THEOREM 15
1.7 THE ENDOMORPHISM TYPE OF A GRAPH 18
1.8 COMMENTS 24
2 GRAPHS AND MATRICES 26
2.1 ADJACENCY MATRIX 26
ISOMORPHIC GRAPHS AND THE ADJACENCY MATRIX 28
COMPONENTS AND THE ADJACENCY MATRIX 29
ADJACENCY LIST 30
2.2 INCIDENCE MATRIX 30
2.3 DISTANCES IN GRAPHS 31
THE ADJACENCY MATRIX AND PATHS 32
THE ADJACENCY MATRIX, THE DISTANCE MATRIX AND CIRCUITS 33
2.4 ENDOMORPHISMS AND COMMUTING GRAPHS 34
2.5 THE CHARACTERISTIC POLYNOMIAL AND EIGENVALUES 35
2.6 CIRCULANT GRAPHS 40
2.7 EIGENVALUES AND THE COMBINATORIAL STRUCTURE 43
COSPECTRAL GRAPHS 43
EIGENVALUES, DIAMETER AND REGULARITY 44
AUTOMORPHISMS AND EIGENVALUES 45
2.8 COMMENTS 46
3 CATEGORIES AND FUNCTORS 48
3.1 CATEGORIES 48
CATEGORIES WITH SETS AND MAPPINGS, I 49
CONSTRUCTS, AND SMALL AND LARGE CATEGORIES 49
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1012312437
DIGITALISIERT DURCH
IMAGE 2
X II CONTENTS
SPECIAL OBJECTS AND MORPHISMS 50
CATEGORIES WITH SETS AND MAPPINGS, II 51
CATEGORIES WITH GRAPHS 51
OTHER CATEGORIES 52
3.2 PRODUCTS & CO 53
COPRODUCTS 53
PRODUCTS 55
TENSOR PRODUCTS 57
CATEGORIES WITH SETS AND MAPPINGS, III 58
3.3 FUNCTORS 58
COVARIANT AND CONTRAVARIANT FUNCTORS 59
COMPOSITION OF FUNCTORS 59
SPECIAL FUNCTORS - EXAMPLES 60
MOR FUNCTORS 60
PROPERTIES OF FUNCTORS 61
3.4 COMMENTS 63
4 BINARY GRAPH OPERATIONS 64
4.1 UNIONS 64
THE UNION 64
THE JOIN 66
THE EDGE SUM 67
4.2 PRODUCTS 70
THE CROSS PRODUCT 71
THE COAMALGAMATED PRODUCT 72
THE DISJUNCTION OF GRAPHS 75
4.3 TENSOR PRODUCTS AND THE PRODUCT IN EGRA 75
THE BOX PRODUCT 76
THE BOXCROSS PRODUCT 79
THE COMPLETE PRODUCT 79
SYNOPSIS OF THE RESULTS 80
PRODUCT CONSTRUCTIONS AS FUNCTORS IN ONE VARIABLE 80
4.4 LEXICOGRAPHIC PRODUCTS AND THE CORONA 81
LEXICOGRAPHIC PRODUCTS 81
THE CORONA 82
4.5 ALGEBRAIC PROPERTIES 83
4.6 MOR CONSTRUCTIONS 85
DIAMOND PRODUCTS 85
LEFT INVERSES FOR TENSOR FUNCTORS 87
POWER PRODUCTS 88
LEFT INVERSES TO PRODUCT FUNCTORS 89
4.7 COMMENTS 90
IMAGE 3
CONTENTS XIII
5 LINE GRAPH AND OTHER UNARY GRAPH OPERATIONS 91
5.1 COMPLEMENTS, OPPOSITE GRAPHS AND GEOMETRIC DUALS 91
5.2 THE LINE GRAPH 92
DETERMINABILITY OF G BY LG 95
5.3 SPECTRA OF LINE GRAPHS 97
WHICH GRAPHS ARE LINE GRAPHS? 99
5.4 THE TOTAL GRAPH 101
5.5 THE TREE GRAPH 102
5.6 COMMENTS 103
6 GRAPHS AND VECTOR SPACES 104
6.1 VERTEX SPACE AND EDGE SPACE 104
THE BOUNDARY & CO 105
MATRIX REPRESENTATION 106
6.2 CYCLE SPACES, BASES & CO 107
THE CYCLE SPACE 107
THE COCYCLE SPACE 109
ORTHOGONALITY I LL
THE BOUNDARY OPERATOR & CO 112
6.3 APPLICATION: MACLANE'S PLANARITY CRITERION 113
6.4 HOMOLOGY OF GRAPHS 116
EXACT SEQUENCES OF VECTOR SPACES 116
CHAIN COMPLEXES AND HOMOLOGY GROUPS OF GRAPHS 117
6.5 APPLICATION: NUMBER OF SPANNING TREES 119
6.6 APPLICATION: ELECTRICAL NETWORKS 123
6.7 APPLICATION: SQUARED RECTANGLES 128
6.8 APPLICATION: SHORTEST (LONGEST) PATHS 132
6.9 COMMENTS 135
7 GRAPHS, GROUPS AND MONOIDS 136
7.1 GROUPS OF A GRAPH 136
EDGE GROUP 137
7.2 ASYMMETRIC GRAPHS AND RIGID GRAPHS 138
7.3 CAYLEY GRAPHS 144
7.4 FRUCHT-TYPE RESULTS 146
FRUCHT'S THEOREM AND ITS GENERALIZATION FOR MONOIDS 147
7.5 GRAPH-THEORETIC REQUIREMENTS 148
SMALLEST GRAPHS FOR GIVEN GROUPS 149
ADDITIONAL PROPERTIES OF GROUP-REALIZING GRAPHS 150
7.6 TRANSFORMATION MONOIDS AND PERMUTATION GROUPS 154
7.7 ACTIONS ON GRAPHS 156
FIXED-POINT-FREE ACTIONS ON GRAPHS 156
IMAGE 4
X IV CONTENTS
TRANSITIVE ACTIONS ON GRAPHS 157
REGULAR ACTIONS 158
7.8 COMMENTS 160
8 THE CHARACTERISTIC POLYNOMIAL OF GRAPHS 161
8.1 EIGENVECTORS OF SYMMETRIC MATRICES 161
EIGENVALUES AND CONNECTEDNESS 162
REGULAR GRAPHS AND EIGENVALUES 163
8.2 INTERPRETATION OF THE COEFFICIENTS OF CHAPO(G) 164
INTERPRETATION OF THE COEFFICIENTS FOR UNDIRECTED GRAPHS 166 8.3 SPECTRA
OF TREES 168
RECURSION FORMULA FOR TREES 168
8.4 THE SPECTRAL RADIUS OF UNDIRECTED GRAPHS 169
SUBGRAPHS 169
UPPER BOUNDS 170
LOWER BOUNDS 171
8.5 SPECTRAL DETERMINABILITY 172
SPECTRAL UNIQUENESS OF K N AND K P Q 172
8.6 EIGENVALUES AND GROUP ACTIONS 174
GROUPS, ORBITS AND EIGENVALUES 174
8.7 TRANSITIVE GRAPHS AND EIGENVALUES 176
DEROGATORY GRAPHS 177
GRAPHS WITH ABELIAN GROUPS 178
8.8 COMMENTS 180
9 GRAPHS AND MONOIDS 181
9.1 SEMIGROUPS 181
9.2 END-REGULAR BIPARTITE GRAPHS 185
REGULAR ENDOMORPHISMS AND RETRACTS 185
END-REGULAR AND END-ORTHODOX CONNECTED BIPARTITE GRAPHS 186 9.3 LOCALLY
STRONG ENDOMORPHISMS OF PATHS 188
UNDIRECTED PATHS 188
DIRECTED PATHS 191
ALGEBRAIC PROPERTIES OF LEND 194
9.4 WREATH PRODUCT OF MONOIDS OVER AN ACT 197
9.5 STRUCTURE OF THE STRONG MONOID 200
THE CANONICAL STRONG DECOMPOSITION OF G 201
DECOMPOSITION OF SEND 202
A GENERALIZED WREATH PRODUCT WITH A SMALL CATEGORY 204
CARDINALITY OF SEND(G) 204
9.6 SOME ALGEBRAIC PROPERTIES OF SEND 205
REGULARITY AND MORE FOR TA 205
IMAGE 5
CONTENTS XV
REGULARITY AND MORE FOR SEND(G) 206
9.7 COMMENTS 207
10 COMPOSITIONS, UNRETRACTIVITIES AND MONOIDS 208
10.1 LEXICOGRAPHIC PRODUCTS 208
10.2 UNRETRACTIVITIES AND LEXICOGRAPHIC PRODUCTS 210
10.3 MONOIDS AND LEXICOGRAPHIC PRODUCTS 214
10.4 THE UNION AND THE JOIN 217
THE SUM OF MONOIDS 217
THE SUM OF ENDOMORPHISM MONOIDS 218
UNRETRACTIVITIES 219
10.5 THE BOX PRODUCT AND THE CROSS PRODUCT 221
UNRETRACTIVITIES 222
THE PRODUCT OF ENDOMORPHISM MONOIDS 223
10.6 COMMENTS 224
11 CAYLEY GRAPHS OF SEMIGROUPS 225
11.1 THE CAY FUNCTOR 225
REFLECTION AND PRESERVATION OF MORPHISMS 227
DOES CAY PRODUCE STRONG HOMOMORPHISMS? 228
11.2 PRODUCTS AND EQUALIZERS 229
CATEGORICAL PRODUCTS 229
EQUALIZERS 231
OTHER PRODUCT CONSTRUCTIONS 232
11.3 CAYLEY GRAPHS OF RIGHT AND LEFT GROUPS 234
11.4 CAYLEY GRAPHS OF STRONG SEMILATTICES OF SEMIGROUPS 237
11.5 APPLICATION: STRONG SEMILATTICES OF (RIGHT OR LEFT) GROUPS 240 11.6
COMMENTS 244
12 VERTEX TRANSITIVE CAYLEY GRAPHS 245
12.1 AUT- VERTEX TRANSITIVITY 245
12.2 APPLICATION TO STRONG SEMILATTICES OF RIGHT GROUPS 247
COLAUT(5, C)-VERTEX TRANSITIVITY 249
AUT(5, C)-VERTEX TRANSITIVITY 250
12.3 APPLICATION TO STRONG SEMILATTICES OF LEFT GROUPS 253
APPLICATION TO STRONG SEMILATTICES OF GROUPS 256
12.4 END'(S, C)-VERTEX TRANSITIVE CAYLEY GRAPHS 256
12.5 COMMENTS 260
13 EMBEDDINGS OF CAYLEY GRAPHS - GENUS OF SEMIGROUPS 261
13.1 THE GENUS OF A GROUP 261
13.2 TOROIDAL RIGHT GROUPS 265
IMAGE 6
XVI CONTENTS
13.3 THE GENUS OF (A X R R ) 270
CAYLEY GRAPHS OF A X R 4 270
CONSTRUCTIONS OF CAYLEY GRAPHS FOR A X R 2 AND A X R3 270
13.4 NON-PLANAR CLIFFORD SEMIGROUPS 275
13.5 PLANAR CLIFFORD SEMIGROUPS 279
13.6 COMMENTS 284
BIBLIOGRAPHY 285
INDEX 301
INDEX OF SYMBOLS 307 |
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.5 |
| dewey-search | 511.5 |
| dewey-sort | 3511.5 |
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| discipline | Mathematik |
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| indexdate | 2024-07-21T00:07:57Z |
| institution | BVB |
| isbn | 3110254085 9783110254082 |
| language | English |
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| physical | XVI, 308 S. graph. Darst. 240 mm x 170 mm |
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| series2 | De Gruyter studies in mathematics |
| spelling | Knauer, Ulrich 1942- Verfasser (DE-588)14404787X aut Algebraic graph theory morphisms, monoids and matrices Ulrich Knauer Berlin [u.a.] De Gruyter 2011 XVI, 308 S. graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 41 Graphentheorie (DE-588)4113782-6 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Graphentheorie (DE-588)4113782-6 s Algebra (DE-588)4001156-2 s DE-604 Erscheint auch als Online-Ausgabe 978-3-11-025509-6 De Gruyter studies in mathematics 41 (DE-604)BV000005407 41 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3832578&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024399130&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Knauer, Ulrich 1942- Algebraic graph theory morphisms, monoids and matrices De Gruyter studies in mathematics Graphentheorie (DE-588)4113782-6 gnd Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4113782-6 (DE-588)4001156-2 (DE-588)4123623-3 |
| title | Algebraic graph theory morphisms, monoids and matrices |
| title_auth | Algebraic graph theory morphisms, monoids and matrices |
| title_exact_search | Algebraic graph theory morphisms, monoids and matrices |
| title_full | Algebraic graph theory morphisms, monoids and matrices Ulrich Knauer |
| title_fullStr | Algebraic graph theory morphisms, monoids and matrices Ulrich Knauer |
| title_full_unstemmed | Algebraic graph theory morphisms, monoids and matrices Ulrich Knauer |
| title_short | Algebraic graph theory |
| title_sort | algebraic graph theory morphisms monoids and matrices |
| title_sub | morphisms, monoids and matrices |
| topic | Graphentheorie (DE-588)4113782-6 gnd Algebra (DE-588)4001156-2 gnd |
| topic_facet | Graphentheorie Algebra Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3832578&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024399130&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
| work_keys_str_mv | AT knauerulrich algebraicgraphtheorymorphismsmonoidsandmatrices |