Applied graph theory: Graphs and electrical networks
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
North-Holland Publ.
1976
|
| Edition: | 2., rev. ed. |
| Series: | North-Holland series in applied mathematics and mechanics
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 542 S. |
Staff View
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| 245 | 1 | 0 | |a Applied graph theory |b Graphs and electrical networks |
| 250 | |a 2., rev. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b North-Holland Publ. |c 1976 | |
| 300 | |a XVI, 542 S. | ||
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Record in the Search Index
| _version_ | 1804136088532418560 |
|---|---|
| adam_text | CONTENTS
Chapter 1. Basic theory 1
1. Introduction 1
2. Basic concepts of abstract graphs 3
2.1. General definitions 3
2.2. Isomorphism 6
2.3. Connectedness 8
2.4. Rank and nullity 11
2.5. Degrees 12
3. Operations on graphs 13
4. Some important classes of graphs 17
4.1. Planar graphs 17
4.2. Separable and nonseparable graphs 19
4.3. Bipartite graphs 22
5. Directed graphs 23
5.1. Basic concepts 24
5.2. Directed edge sequence 27
5.3. Outgoing and incoming degrees 29
5.4. Strongly connected directed graphs 30
5.5. Some important classes of directed graphs 31
6. Mixed graphs 32
7. Conclusions 32
Problems 33
Chapter 2. Foundations of electrical network theory 36
1. Matrices and directed graphs 37
1.1. The node edge incidence matrix 37
1.2. The circuit edge incidence matrix 41
1.3. The cut edge incidence matrix 46
1.4. Interrelationships among the matrices A, Bf, and Qs 53
1.5. Vector spaces associated with the matrices Ba and Qa 57
2. The electrical network problem 58
3. Solutions of the electrical network problem 62
3.1. Branch current and branch voltage systems of equations 63
3.2. Loop system of equations 63
3.3. Cut system of equations 70
3.4. Additional considerations 76
4. Invariance and mutual relations of network determinants and the generalized
cofactors 77
xiv Contents
4.1. A brief history 77
4.2. Preliminary considerations 78
4.3. The loop and cut transformations 83
4.4. Network matrices 85
4.5. Generalized cofactors of the elements of the network matrix 95
5. Invariance and the incidence functions 107
6. Topological formulas for RLC networks 111
6.1. Network determinants and trees and cotrees 111
6.2. Generalized cofactors and 2 trees and 2 cotrees 114
6.3. Topological formulas for RLC two port networks 122
7. The existence and uniqueness of the network solutions 125
8. Conclusions 132
Problems 133
Chapter 3. Directed graph solutions of linear algebraic equations 140
1. The associated Coates graph 141
1.1. Topological evaluation of determinants 142
1.2. Topological evaluation of cofactors 146
1.3. Topological solutions of linear algebraic equations 149
1.4. Equivalence and transformations 155
2. The associated Mason graph 167
2.1. Topological evaluation of determinants 169
2.2. Topological evaluation of cofactors 172
2.3. Topological solutions of linear algebraic equations 174
2.4. Equivalence and transformations 177
3. The modifications of Coates and Mason graphs 189
3.1. Modifications of Coates graphs 189
3.2. Modifications of Mason graphs 197
4. The generation of subgraphs of a directed graph 199
4.1. The generation of 1 factors and 1 factorial connections 201
4.2. The generation of semifactors and fc semifactors 203
5. The eigenvalue problem 206
6. The matrix inversion 210
7. Conclusions 216
Problems 216
Chapter 4. Topological analysis of linear systems 224
1. The equicofactor matrix 225
2. The associated directed graph 230
2.1. Directed trees and first order cofactors 231
2.2. Directed 2 trees and second order cofactors 244
3. Equivalence and transformations 251
4. The associated directed graph and the Coates graph 262
4.1. Directed trees, 1 factors, and semifactors 262
4.2. Directed 2 trees, 1 factorial connections, and 1 semifactors 266
5. Generation of directed trees and directed 2 trees 269
Contents xv
5.1. Algebraic formulation 269
5.2. Iterative procedure 272
5.3. Partial factoring 279
6. Direct analysis of electrical networks 281
6.1. Open circuit transfer impedance and voltage gain functions 281
6.2. Short circuit transfer admittance and current gain functions 289
6.3. Open circuit impedance and short circuit admittance matrices 294
6.4. The physical significance of the associated directed graph 297
6.5. Direct analysis of the associated directed graph 302
7. Conclusions 311
Problems 312
Chapter 5. Trees and their generation 320
1. The characterizations of a tree 320
2. The codifying of a tree structure 325
2.1. Codification by paths 326
2.2. Codification by terminal edges 328
3. Decomposition into paths 330
4. The Wang algebra formulation 332
4.1. The Wang algebra 333
4.2. Linear dependence 334
4.3. Trees and cotrees 338
4.4. Multi trees and multi cotrees 340
4.5. Decomposition 345
5. Generation of trees by decomposition without duplications 353
5.1. Essential complementary partitions of a set 353
5.2. Algorithm 356
5.3. Decomposition without duplications 359
6. The matrix formulation 365
6.1. The enumeration of major submatrices of an arbitrary matrix 365
6.2. Trees and cotrees 368
6.3. Directed trees and directed 2 trees 370
7. Elementary transformations 373
8. Hamilton circuits in directed tree graphs 379
9. Directed trees and directed Euler lines 384
10. Conclusions 389
Problems 390
Chapter 6. The realizability of directed graphs with prescribed degrees 398
1. Existence and realization as a (p, s) digraph 398
1.1. Directed graphs and directed bipartite graphs 400
1.2. Existence 401
1.3. A simple algorithm for the realization 413
1.4. Degree invariant transformations 419
1.5. Realizability as a connected (j , i) digraph 422
2. Realizability as a symmetric (p, i) digraph 427
xvi Contents
2.1. Existence 428
2.2. Realization 433
2.3. Realizability as connected, separable and nonseparable graphs 436
3. Unique realizability of graphs without self loops 440
3.1. Preliminary considerations 441
3.2. Unique realizability as a connected graph 443
3.3. Unique realizability as a graph 446
4. Existence and realization of a (p, i ) matrix 448
5. Realizability as a weighted directed graph 452
6. Conclusions 454
Problems 455
Chapter 7. State equations of networks 464
1. State equations in normal form 464
2. Procedures for writing the state equations 472
3. The explicit form of the state equation 479
4. An alternative representation of the state equation 490
5. Physical interpretations of the parameter matrices 491
6. Order of complexity 499
6.1 Relations between det H(s) and network determinant 504
6.2 RLC networks 508
6.3 Active networks 511
7. Conclusions 514
Problems 515
Bibliography 518
Symbol index 529
Subject index 534
|
| adam_txt |
CONTENTS
Chapter 1. Basic theory 1
1. Introduction 1
2. Basic concepts of abstract graphs 3
2.1. General definitions 3
2.2. Isomorphism 6
2.3. Connectedness 8
2.4. Rank and nullity 11
2.5. Degrees 12
3. Operations on graphs 13
4. Some important classes of graphs 17
4.1. Planar graphs 17
4.2. Separable and nonseparable graphs 19
4.3. Bipartite graphs 22
5. Directed graphs 23
5.1. Basic concepts 24
5.2. Directed edge sequence 27
5.3. Outgoing and incoming degrees 29
5.4. Strongly connected directed graphs 30
5.5. Some important classes of directed graphs 31
6. Mixed graphs 32
7. Conclusions 32
Problems 33
Chapter 2. Foundations of electrical network theory 36
1. Matrices and directed graphs 37
1.1. The node edge incidence matrix 37
1.2. The circuit edge incidence matrix 41
1.3. The cut edge incidence matrix 46
1.4. Interrelationships among the matrices A, Bf, and Qs 53
1.5. Vector spaces associated with the matrices Ba and Qa 57
2. The electrical network problem 58
3. Solutions of the electrical network problem 62
3.1. Branch current and branch voltage systems of equations 63
3.2. Loop system of equations 63
3.3. Cut system of equations 70
3.4. Additional considerations 76
4. Invariance and mutual relations of network determinants and the generalized
cofactors 77
xiv Contents
4.1. A brief history 77
4.2. Preliminary considerations 78
4.3. The loop and cut transformations 83
4.4. Network matrices 85
4.5. Generalized cofactors of the elements of the network matrix 95
5. Invariance and the incidence functions 107
6. Topological formulas for RLC networks 111
6.1. Network determinants and trees and cotrees 111
6.2. Generalized cofactors and 2 trees and 2 cotrees 114
6.3. Topological formulas for RLC two port networks 122
7. The existence and uniqueness of the network solutions 125
8. Conclusions 132
Problems 133
Chapter 3. Directed graph solutions of linear algebraic equations 140
1. The associated Coates graph 141
1.1. Topological evaluation of determinants 142
1.2. Topological evaluation of cofactors 146
1.3. Topological solutions of linear algebraic equations 149
1.4. Equivalence and transformations 155
2. The associated Mason graph 167
2.1. Topological evaluation of determinants 169
2.2. Topological evaluation of cofactors 172
2.3. Topological solutions of linear algebraic equations 174
2.4. Equivalence and transformations 177
3. The modifications of Coates and Mason graphs 189
3.1. Modifications of Coates graphs 189
3.2. Modifications of Mason graphs 197
4. The generation of subgraphs of a directed graph 199
4.1. The generation of 1 factors and 1 factorial connections 201
4.2. The generation of semifactors and fc semifactors 203
5. The eigenvalue problem 206
6. The matrix inversion 210
7. Conclusions 216
Problems 216
Chapter 4. Topological analysis of linear systems 224
1. The equicofactor matrix 225
2. The associated directed graph 230
2.1. Directed trees and first order cofactors 231
2.2. Directed 2 trees and second order cofactors 244
3. Equivalence and transformations 251
4. The associated directed graph and the Coates graph 262
4.1. Directed trees, 1 factors, and semifactors 262
4.2. Directed 2 trees, 1 factorial connections, and 1 semifactors 266
5. Generation of directed trees and directed 2 trees 269
Contents xv
5.1. Algebraic formulation 269
5.2. Iterative procedure 272
5.3. Partial factoring 279
6. Direct analysis of electrical networks 281
6.1. Open circuit transfer impedance and voltage gain functions 281
6.2. Short circuit transfer admittance and current gain functions 289
6.3. Open circuit impedance and short circuit admittance matrices 294
6.4. The physical significance of the associated directed graph 297
6.5. Direct analysis of the associated directed graph 302
7. Conclusions 311
Problems 312
Chapter 5. Trees and their generation 320
1. The characterizations of a tree 320
2. The codifying of a tree structure 325
2.1. Codification by paths 326
2.2. Codification by terminal edges 328
3. Decomposition into paths 330
4. The Wang algebra formulation 332
4.1. The Wang algebra 333
4.2. Linear dependence 334
4.3. Trees and cotrees 338
4.4. Multi trees and multi cotrees 340
4.5. Decomposition 345
5. Generation of trees by decomposition without duplications 353
5.1. Essential complementary partitions of a set 353
5.2. Algorithm 356
5.3. Decomposition without duplications 359
6. The matrix formulation 365
6.1. The enumeration of major submatrices of an arbitrary matrix 365
6.2. Trees and cotrees 368
6.3. Directed trees and directed 2 trees 370
7. Elementary transformations 373
8. Hamilton circuits in directed tree graphs 379
9. Directed trees and directed Euler lines 384
10. Conclusions 389
Problems 390
Chapter 6. The realizability of directed graphs with prescribed degrees 398
1. Existence and realization as a (p, s) digraph 398
1.1. Directed graphs and directed bipartite graphs 400
1.2. Existence 401
1.3. A simple algorithm for the realization 413
1.4. Degree invariant transformations 419
1.5. Realizability as a connected (j , i) digraph 422
2. Realizability as a symmetric (p, i) digraph 427
xvi Contents
2.1. Existence 428
2.2. Realization 433
2.3. Realizability as connected, separable and nonseparable graphs 436
3. Unique realizability of graphs without self loops 440
3.1. Preliminary considerations 441
3.2. Unique realizability as a connected graph 443
3.3. Unique realizability as a graph 446
4. Existence and realization of a (p, i ) matrix 448
5. Realizability as a weighted directed graph 452
6. Conclusions 454
Problems 455
Chapter 7. State equations of networks 464
1. State equations in normal form 464
2. Procedures for writing the state equations 472
3. The explicit form of the state equation 479
4. An alternative representation of the state equation 490
5. Physical interpretations of the parameter matrices 491
6. Order of complexity 499
6.1 Relations between det H(s) and network determinant 504
6.2 RLC networks 508
6.3 Active networks 511
7. Conclusions 514
Problems 515
Bibliography 518
Symbol index 529
Subject index 534 |
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| indexdate | 2024-07-09T20:50:50Z |
| institution | BVB |
| language | English |
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| physical | XVI, 542 S. |
| publishDate | 1976 |
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| series | North-Holland series in applied mathematics and mechanics |
| series2 | North-Holland series in applied mathematics and mechanics |
| spelling | Zhen, Waigai Verfasser aut Applied graph theory Graphs and electrical networks 2., rev. ed. Amsterdam [u.a.] North-Holland Publ. 1976 XVI, 542 S. txt rdacontent n rdamedia nc rdacarrier North-Holland series in applied mathematics and mechanics 13 Elektrisches Netz (DE-588)4138980-3 gnd rswk-swf Graph (DE-588)4021842-9 gnd rswk-swf Graph (DE-588)4021842-9 s DE-604 Elektrisches Netz (DE-588)4138980-3 s North-Holland series in applied mathematics and mechanics 13 (DE-604)BV001900898 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015329094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zhen, Waigai Applied graph theory Graphs and electrical networks North-Holland series in applied mathematics and mechanics Elektrisches Netz (DE-588)4138980-3 gnd Graph (DE-588)4021842-9 gnd |
| subject_GND | (DE-588)4138980-3 (DE-588)4021842-9 |
| title | Applied graph theory Graphs and electrical networks |
| title_auth | Applied graph theory Graphs and electrical networks |
| title_exact_search | Applied graph theory Graphs and electrical networks |
| title_exact_search_txtP | Applied graph theory Graphs and electrical networks |
| title_full | Applied graph theory Graphs and electrical networks |
| title_fullStr | Applied graph theory Graphs and electrical networks |
| title_full_unstemmed | Applied graph theory Graphs and electrical networks |
| title_short | Applied graph theory |
| title_sort | applied graph theory graphs and electrical networks |
| title_sub | Graphs and electrical networks |
| topic | Elektrisches Netz (DE-588)4138980-3 gnd Graph (DE-588)4021842-9 gnd |
| topic_facet | Elektrisches Netz Graph |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015329094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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