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Algebraic Graph Theory: Morphisms, Monoids and Matrices<...

Algebraic Graph Theory: Morphisms, Monoids and MatricesTitoloAlgebraic Graph Theory: Morphisms, Monoids and Matrices
AutoreKnauer, U.
Prezzo
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€ 101,23   Spedizioni gratuite in Italia
(Prezzo € 103,30)
CategoriaMathematics: Algebra
Mathematics: Group Theory
Mathematics: Matrices
RilegaturaHardcover
Età consigliata22 anni
Dati308 p.
Anno2011
EditoreWalter de Gruyter
Collanade Gruyter Studies in Mathematics
Normalmente disponibile per la spedizione entro 4/5 settimane

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Indice e argomenti trattati
Prefacev
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
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
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
111
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
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 Kn and Kp, 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
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(S, C)-vertex transitivity
249
Aut(S, 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
13.3 The genus of (A x Rr)
270
Cayley graphs of A x R4
270
Constructions of Cayley graphs for A x R2 and A x R3
270
13.4 Non-planar Clifford semigroups
275
13.5 Planar Clifford semigroups
279
13.6 Comments
284
Bibliography285
Index301
Index of symbols307

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