Brualdi R.A., Ryser H.J. — Combinatorial Matrix Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Brualdi R.A., Ryser H.J. — Combinatorial Matrix Theory

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Название: Combinatorial Matrix Theory

Авторы: Brualdi R.A., Ryser H.J.

Аннотация:

The book deals with the many connections between matrices, graphs, diagraphs and bipartite graphs. The basic theory of network flows is developed in order to obtain existence theorems for matrices with prescribed combinatorical properties and to obtain various matrix decomposition theorems. Other chapters cover the permanent of a matrix and Latin squares. The book ends by considering algebraic characterizations of combinatorical properties and the use of combinatorial arguments in proving classical algebraic theorems, including the Cayley-Hamilton Theorem and the Jorda Canonical Form.

Язык:

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 380

Добавлена в каталог: 16.05.2011

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Предметный указатель

${e}_{n}$       145
${l}_{2}$ -norm      213—214
Acyclic digraph      335 340 343
Adjacency matrix      25 48—52 53 179 317
Adjacency matrix, formal      317—323 324
Adjacency matrix, isomorphic      157
Adjacency matrix, regular      157 160
Adjacency matrix, tree      343
Adjugate      301
Admittance matrix      30 see
Affine plane      278—80
Affine plane, order of      280
Amitsur — Levitzki theorem      331—334
ARC      53
Arc, back arc      98
Arc, capacity of      164
Arc, cross arc      98
Arc, dominant      89—90
Arc, forest arc      98
Arc, forward arc      98
Arc, weight of      291—292
Bipartite graph      44 107—144 178 317
Bipartite graph, adjacency matrix of      44
Bipartite graph, bipartition of      44 107
Bipartite graph, complete      108 135 147 148 162:
Bipartite graph, complete, line graph of      153
Bipartite graph, elementary      124
Birkhoffs theorem      9
BLOCKS      278
Branch      62 63
Cayley table      250—251 254—255 258 262 268 271
Cayley — Hamilton theorem      327—329 333
Chain      25 210
Chain, directed      54
Chain, length of      210
Chain, weight of      211 212
Characteristic polynomial      26 28 36 76—77 88 328—329
Chromatic index      51
Circuit      54
Circulant matrix      158—162 233
Circulation      168—169
Co-term rank      125—126 134
Cocktail party graph      36—37
Column-linear set      139
Complete digraph      291
Complete graph      23 26 39 318
Complete graph, line graph of      152
Complete graph, spectrum of      28
Complete mapping      251
Completely reducible matrix      76
Complexity (of a graph)      38—40 324
composition      191 196
Conference graph      151 155
Conference matrix      154
Configuration      3 20 293 306 307—308
Configuration, complementary      305
Configuration, dual      4 12
Configuration, isomorphic      4
Conjugate partition      193 340
Connectivity, algebraic      40—53
Connectivity, edge      41
Connectivity, vertex      41
Contraction      62 242
Convex closure      193—195
Convex sequence      192 340
Cospectral graphs      26—27 155
Cover (of digraph)      134
Covering sequence      190—196
Cut      166
Cut, capacity of      166
CYCLE      25
Cycle graph      28
Cycle hypergraph      144
Cycle, directed      54
Cyclic components      71
Cyclic components, computation of      103—105
Cyclic matrix      77
Decision problem      245—248
Decision problem, polynomial reducible      246
Decision problem, size of      245
Decomposition      251
Decomposition theorems      108 125—136 184—188
Degree of vertex      24
demand      169
Depth-first number      97
Depth-first search      97
Derangement number Dk      17 222—223 233
Derangements      201—202 285
Determinant      8 15 16 21—22 76 89 90 92 93 94 95 134 209—214 235—48 291—293 294 295 296—297 298—299 301 309—310 311 313—314 314—316 319 323 325—326
Determinantal divisor      337
Diagonal      112
Diagonal structure (hypergraph)      136—144
Diagonal structure (hypergraph), isomorphism of      136—137
Diagonal structure (hypergraph), partial transposition      142—143
Diagonal structure (hypergraph), permutation      140
Diagonal structure (hypergraph), transposition      140
Difference sequence      191 195—196 338 339—340
Digraph (directed graph)      53 243
Digraph (directed graph), adjacency matrix of      53
Digraph (directed graph), capacity function      164
Digraph (directed graph), characteristic polynomial      88
Digraph (directed graph), condensation      56
Digraph (directed graph), cyclically r-partite      70
Digraph (directed graph), general      53
Digraph (directed graph), girth of      93
Digraph (directed graph), imprimitive      68—69
Digraph (directed graph), indegree sequence      173 176
Digraph (directed graph), minimally strong      61—68
Digraph (directed graph), out degree sequence      173 176
Digraph (directed graph), primitive      68
Digraph (directed graph), regular      53
Digraph (directed graph), spanning subdigraph      97
Digraph (directed graph), spectrum      88
Digraph (directed graph), splitting of      244
Digraph (directed graph), strongly connected      54 55—61
Digraph (directed graph), vertex-weighted      89—90
Directed cycle, dominant      90
Directed cycle, weight of      243
Directed forest      96—99
Directed forest, spanning      97
Divisor sequence      337—338 339—340
Doubly stochastic matrix      9 11 117
EDGE      23
Edge coloring      50
Edge, endpoints      23
Eigenvalue inclusion regions      89—95
Eigenvalues      16 26 27 76—77 88—96 130—132 145 146
Elementary divisors      337
Elementary similarity      341
Essential column      110
Essential line      110
Essential row      110
Euler $\phi$ -function      21—22
Euler conjecture      275 282—283
Eulerian matrix      31
Eulerian trail      332
Evans conjecture      263—267
Even digraph      243—244
EXPONENT      78—87 83—85 117
Exponent, gaps      84
Factor      48
Ferrers matrix      206—207 208 217
Flow (network)      164—171
Flow (network), supply-demand      169—171
Flow (network), value of      165
Forest      178
Friendship theorem      151
Frobenius normal form      58 96—102
Frobenius normal form, computation of      96—106
Frobenius — Schur index      72 81

Fully indecomposable components      115 140—141 297
Fully indecomposable matrix      110—118 124 228—233 296—297 299 303
Fully indecomposable matrix, inductive structure      116
Fundamental trace identity      327 329—331
Gale — Ryser Theorem      176
Galois affine plane      279 280
General digraph      53
Generalized matrix function      295
Generic matrix      294—297
Generic nilpotent matrix      335—343
Generic skew-symmetric matric      317
Gersgorin's theorem      88
Graph (general)      23 24
Graph (general), adjacent      23
Graph (general), complement      23
Graph (general), complete      23 26
Graph (general), complete multipartite      147 155
Graph (general), connected      27
Graph (general), connected components      27
Graph (general), cospectral      26—27 155
Graph (general), cubic      24 37
Graph (general), decomposition of      108 109
Graph (general), degree sequence      179 183
Graph (general), diameter      27
Graph (general), disconnected      27
Graph (general), distance      27
Graph (general), edge      23
Graph (general), induced      23
Graph (general), isolated      23
Graph (general), isomorphic      23 25 26
Graph (general), multiplicity of      24
Graph (general), order      23
Graph (general), regular      24 37 43
Graph (general), simple      23
Graph (general), spanning      23 30
Graph (general), spectrum of      28
Graph (general), subgraph      23
Graph (general), vertices      23
Hadamard product      294 317
Hadamard's determinant inequality      214
Hall's theorem      299—301
Hamilton cycle problem      245—246 247
Helly type theorem      19—20
Hoffman polynomial      147 157 162
Hoffman — Singleton graph      154
Horizontal line      277
Imprimitive matrix      70
Incidence matrix      3 29 35 230 304 306
Incidence matrix, formal      293—303
Incidence matrix, oriented      29 38
Incidence matrix, weighted      210—214
INDEX      103—105
Index of imprimitivity      68—78
Index of imprimitivity, computation of      103—105
Integer sequences      191—195
Intersection matrix      11—12
Intersection matrix, formal      304—310
Intersection matrix, symmetric      305 306 309
Invariant factors      337
Irreducible components      58
Irreducible components, computation of      96—106
Irreducible matrix      55—61 113 145 297—300
Irreducible matrix, inductive structure      67—68
Jacobi's identity      321—322 301
Join      192
Jordan block      336 340—343
Jordan canonical form      335—337 340—343
Jordan partition      336 338 339—340 343
Jordan partition, length of      336
K-decomposition      187
k-path      336—337 338
k-path number      336—337
k-subset      203
Koenig's Theorem      6 9 44 47 125 128 196 300 301—303
Laplacian matrix      30 38 41 325
Laplacian matrix, formal      324—327
Latin rectangle      250—253
Latin rectangle, completion of      260—262 68
Latin square      250—290
Latin square, completion of      259—262
Latin square, embedding of      262—263
Latin square, enlargement of      265—267
Latin square, equivalent      253
Latin square, horizontal line of      270
Latin square, idempotent      281—283 283—284
Latin square, latin line of      270 277
Latin square, mutually orthogonal      272—275 286—287
Latin square, normalized      250
Latin square, number of      284—285
Latin square, partial      259—269
Latin square, partitioning of      262—263
Latin square, self-orthogonal      287—288 289
Latin square, symmetric      251 268 269 283
Latin square, vertical line of      270
Lattice graph      153
Line      1 252 277 278
Line cover      6 110—112
Line graph      35—37
Line graph, spectrum of      35
Linear set      139
Linearizable set      143 144
Lowlink      100—101
MacMahon's master theorem      310—316
MacNeish's conjecture      275 282—283
Majorization      175
Matching      44—52 112 189
Matching sequence      189—196
Matching, in digraph      134
Matching, perfect      48
Matrix      1
Matrix, back diagonal of      263
Matrix, column sum vector      172
Matrix, complement of      3 12
Matrix, cover of      6
Matrix, cyclic      70
Matrix, cyclic components      71
Matrix, diagonally dominant      89
Matrix, elementwise product      236
Matrix, existence theorems for      172—184
Matrix, line of      1
Matrix, number of l's      123
Matrix, permutation      1
Matrix, principal      17
Matrix, proper cover of      6
Matrix, row sum vector      172
Matrix, submatrix of      17
Matrix, sum of elements of      185
Matrix, triangle of      18
Matrix, upper triangular      338—339
Matrix-tree theorem      325—326
Matroid, regular      31—34
Matroid, unimodular      31—34
Max flow-mincut theorem      166—168
Meet      192
Menage numbers      202—203 204—205 233 234 285
Minimax theorem      6
Mixed matrix      303
Moore graph      153
Moore graph, generalized      154
Multicolored graph      131—132
Multiedge      24
Multigraph      23
n-set      3
Nearly decomposable matrix      118—124 228
Nearly decomposable matrix, inductive structure      120—121
Nearly decomposable matrix, number of l's of      122
Nearly reducible matrix      61—68 118 124
Nearly reducible matrix, inductive structure      64
Nearly reducible matrix, number of, 1's of      66
Net      277—280

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