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E-raamat: Graph Theory And Its Engineering Applications

(Univ Of Illinois, Chicago, Usa)
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Graphs are extremely useful in modeling systems in physical sciences and engineering problems, because of their intuitive diagrammatic nature. This text gives a reasonably deep account of material closely related to engineering applications. Topics like directed-graph solutions of linear equations, topological analysis of linear systems, state equations, rectangle dissection and layouts, and network flows are included. A major theme of the book is electrical network theory.This book is basically intended as a reference text for researchers, and requires a certain level of mathematical maturity. However the text may equally well be used for graduate level courses on network topology and linear systems and circuits. Some of the later chapters are suitable as topics for advanced seminars. A special feature of the book is that references to other published literature are included for almost all the results presented, making the book especially handy for those wishing to continue with a study of special topics.
Preface vii
CHAPTER
1. Basic theory
1(35)
1. Introduction
1(2)
2. Basic concepts of abstract graphs
3(10)
2.1. General definitions
3(3)
2.2. Isomorphism
6(2)
2.3. Connectedness
8(3)
2.4. Rank and nullity
11(1)
2.5. Degrees
12(1)
3. Operations on graphs
13(4)
4. Some important classes of graphs
17(6)
4.1. Planar graphs
17(2)
4.2. Separable and nonseparable graphs
19(3)
4.3. Bipartite graphs
22(1)
5. Directed graphs
23(9)
5.1. Basic concepts
24(3)
5.2. Directed-edge sequence
27(2)
5.3. Outgoing and incoming degrees
29(1)
5.4. Strongly-connected directed graphs
30(1)
5.5. Some important classes of directed graphs
31(1)
6. Mixed graphs
32(1)
7. Conclusions
32(1)
8. Problems
33(3)
CHAPTER
2. Foundations of electrical network theory
36(104)
1. Matrices and directed graphs
37(21)
1.1. The node-edge incidence matrix
37(4)
1.2. The circuit-edge incidence matrix
41(5)
1.3. The cut-edge incidence matrix
46(7)
1.4. Interrelationships among the matrices A, Bf, and Qf
53(4)
1.5. Vector spaces associated with the matrices Ba and Qa
57(1)
2. The electrical network problem
58(4)
3. Solutions of the electrical network problem
62(15)
3.1. Branch-current and branch-voltage systems of equations
63(1)
3.2. Loop system of equations
63(7)
3.3. Cut system of equations
70(6)
3.4. Additional considerations
76(1)
4. Invariance and mutual relations of network determinants and the generalized cofactors
77(30)
4.1. A brief history
77(1)
4.2. Preliminary considerations
78(5)
4.3. The loop and cut transformations
83(2)
4.4. Network matrices
85(10)
4.5. Generalized cofactors of the elements of the network matrix
95(12)
5. Invariance and the incidence functions
107(4)
6. Topological formulas for RLC networks
111(14)
6.1. Network determinants and trees and cotrees
111(3)
6.2. Generalized cofactors and 2-trees and 2-cotrees
114(8)
6.3. Topological formulas for RLC two-port networks
122(3)
7. The existence and uniqueness of the network solutions
125(7)
8. Conclusions
132(1)
Problems
133(7)
CHAPTER
3. Directed-graph solutions of linear algebraic equations
140(84)
1. The associated Coates graph
141(26)
1.1. Topological evaluation of determinants
142(4)
1.2. Topological evaluation of cofactors
146(3)
1.3. Topological solutions of linear algebraic equations
149(6)
1.4. Equivalence and transformations
155(12)
2. The associated Mason graph
167(20)
2.1. Topological evaluation of the determinants
169(3)
2.2. Topological evaluation of cofactors
172(2)
2.3. Topological solutions of linear algebraic equations
174(3)
2.4. Equivalence and transformations
177(12)
3. The modifications of Coates and Mason graphs
189(10)
3.1. Modifications of Coates graphs
189(8)
3.2. Modifications of Mason graphs
197(2)
4. The generation of subgraphs of a directed graph
199(7)
4.1. The generation of 1-factors and 1-factorial connections
201(2)
4.2. The generation of semifactors and k-semifactors
203(3)
5. The eigenvalue problem
206(4)
6. The matrix inversion
210(6)
7. Conclusions
216(1)
8. Problems
216(8)
CHAPTER
4. Topological analysis of linear systems
224(96)
1. The equicofactor matrix
225(5)
2. The associated directed graph
230(21)
2.1. Directed-trees and first-order cofactors
231(13)
2.2. Directed 2-trees and second-order cofactors
244(7)
3. Equivalence and transformations
251(11)
4. The associated directed graph and the Coates graph
262(7)
4.1. Directed trees, 1-factors, and semifactors
262(4)
4.2. Directed 2-trees, 1-factorial connections, and 1-semifactors
266(3)
5. Generation of directed trees and directed 2-trees
269(12)
5.1. Algebraic formulation
269(3)
5.2. Iterative procedure
272(7)
5.3. Partial factoring
279(2)
6. Direct analysis of electrical networks
281(30)
6.1. Open-circuit transfer-impedance and voltage-gain functions
281(8)
6.2. Short-circuit transfer-admittance and current-gain functions
289(5)
6.3. Open-circuit impedance and short-circuit admittance matrices
294(3)
6.4. The physical significance of the associated directed graph
297(5)
6.5. Direct analysis of the associated directed graph
302(9)
7. Conclusions
311(1)
Problems
312(8)
CHAPTER
5. Trees and their generation
320(78)
1. The characterizations of a tree
320(5)
2. The codifying of a tree-structure
325(5)
2.1. Codification by paths
326(2)
2.2. Codification by terminal edges
328(2)
3. Decomposition into paths
330(2)
4. The Wang-algebra formulation
332(21)
4.1. The Wang algebra
333(1)
4.2. Linear dependence
334(4)
4.3. Trees and cotrees
338(2)
4.4. Multi-trees and multi-cotrees
340(5)
4.5. Decomposition
345(8)
5. Generation of trees by decomposition without duplications
353(12)
5.1. Essential complementary partitions of a set
353(3)
5.2. Algorithm
356(3)
5.3. Decomposition without duplications
359(6)
6. The matrix formulation
365(8)
6.1. The enumeration of major submatrices of an arbitrary matrix
365(3)
6.2. Trees and cotrees
368(2)
6.3. Directed trees and directed 2-trees
370(3)
7. Elementary transformations
373(6)
8. Hamilton circuits in directed-tree graphs
379(5)
9. Directed trees and directed Euler lines
384(5)
10. Conclusions
389(1)
Problems
390(8)
CHAPTER
6. The realizability of directed graphs with prescribed degrees
398(66)
1. Existence and realization as a (p, s)-digraph
398(29)
1.1 Directed graphs and directed bipartite graphs
400(1)
1.2. Existence
401(12)
1.3. A simple algorithm for the realization
413(6)
1.4. Degree invariant transformations
419(3)
1.5. Realizability as a connected (p, s)-digraph
422(5)
2. Realizability as a symmetric (p, s)-digraph
427(13)
2.1. Existence
428(5)
2.2. Realization
433(3)
2.3. Realizability as connected, separable and nonseparable graphs
436(4)
3. Unique realizability of graphs without self-loops
440(8)
3.1. Preliminary considerations
441(2)
3.2. Unique realizability as a connected graph
443(3)
3.3. Unique realizability as a graph
446(2)
4. Existence and realization of a (p, s)-matrix
448(4)
5. Realizability as a weighted directed graph
452(2)
6. Conclusions
454(1)
Problems
455(9)
CHAPTER
7. State equations of networks
464(54)
1. State equations in normal form
464(8)
2. Procedures for writing the state equations
472(8)
3. The explicit form of the state equation
480(10)
4. An alternative representation of the state equation
490(1)
5. Physical interpretations of the parameter matrices
491(8)
6. Order of complexity
499(15)
6.1. Relations between det H(s) and network determinant
504(4)
6.2. RLC networks
508(3)
6.3 Active networks
511(3)
7. Conclusions
514(1)
Problems
515(3)
CHAPTER
8. Squaring rectangles and layouts
518(94)
1. Introduction
518(3)
2. The Bouwkamp code
521(3)
3. The electrical network associated with a dissected rectangle
524(11)
4. Characterization of the c-nets and c-digraphs
535(6)
5. Perfect subdivison of the general rectangle
541(12)
5.1. The perfect rectangles Rn
542(2)
5.2. The perfect square Sn
544(7)
5.3. Sequence of perfect squares
551(2)
6. Extension to perfect rectangular parallelepiped
553(1)
7. VLSI layout
554(51)
7.1. The zero wasted area floorplan with continuous aspect ratios
557(17)
7.2. Solving the nonlinear nodal equations
574(7)
7.3. Floorplan area optimization with constrained aspect ratio
581(15)
7.4. From digraphs to layout
596(2)
7.5. Graph-theoretic characterization of the minimum area layout
598(7)
8. Conclusions
605(2)
Problem
607(5)
CHAPTER
9. Graphical matrices
612(59)
1. Totally unimodular matrix
612(16)
2. Regular and binary matrices
628(11)
3. Circuit and cutset matrices
639(14)
4. 2-isomorphism
653(3)
5. Graph realization
656(8)
6. Conclusions
664(1)
Problems
665(6)
Bibliography 671(13)
Symbol index 684(6)
Subject index 690
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