Linear Graphs and Electrical Networks

Title page......Page 1
Preface......Page 5
Contents......Page 7
1-1 Survey of applications......Page 9
1-2 The nonoriented graph......Page 16
Problems......Page 25
2-1 The Königsberger bridge problem......Page 27
2-2 Circuits......Page 30
2-3 Trees and fundamental systems of circuits......Page 32
2-4 Cut-sets and fundamental systems of cut-sets......Page 36
2-5 Cut-sets and circuits......Page 40
Problems......Page 41
3-1 Nonseparable graphs......Page 43
3-2 Planar graphs......Page 47
3-3 Dual graphs......Page 49
3-4 One terminal-pair graphs......Page 59
Problems......Page 61
4-1 The field modulo......Page 63
4-2 The vertex or incidence matrix......Page 69
4-3 The circuit matrix......Page 72
4-4 Nonsingular submatrices of A and B and formula for B_I......Page 76
4-5 The cut-set matrix......Page 80
4-6 Linear vector spaces......Page 84
4-7 Vector spaces associated with a graph......Page 91
Problems......Page 94
5-1 The vertex matrix......Page 96
5-2 The circuit matrix......Page 98
5-3 Nonsingular submatrices of A and B and formula for B_I......Page 100
5-4 Cut-sets of directed graphs......Page 103
5-5 Existence of graphs for given matrices......Page 106
5-6 Summary of important results on graphs......Page 119
Problems......Page 122
6-1 Kirchhoff's laws......Page 125
6-2 Mesh (loop) and node transformations......Page 130
6-3 The third postulate......Page 135
6-4 Loop and node systems of equations......Page 140
6-5 Energy functions and stability......Page 152
6-6 Dual networks......Page 157
Problems......Page 161
7-1 Node determinant and cofactors......Page 163
7-2 Driving-point and transfer admittances......Page 173
7-3 The short-circuit admittance functions......Page 176
7-4 Kirchhoff's rules......Page 179
7-5 General linear networks......Page 186
Problems......Page 202
8-1 Enumeration of natural frequencies......Page 205
8-2 One terminal-pair networks......Page 208
8-3 Two terminal-pair networks......Page 216
Problems......Page 234
9-1 Contact networks......Page 235
9-2 Sequential machines......Page 258
9-3 Logic networks......Page 268
Problems......Page 273
10-1 Communication networks......Page 276
10-2 Flow graphs and signal-flow graphs......Page 281
10-3 Calculus of binary relations......Page 294
10-4 Logic: axiomatics......Page 295
10-5 Brief survey of other applications......Page 297
Problems......Page 299
RESEARCH PROBLEMS......Page 303
BIBLIOGRAPHY......Page 313
INDEX......Page 325