# Applied Graph Theory, Volume 13

## 2nd Edition

### Graphs and Electrical Networks

**Editors:**H. A. Lauwerier W. T. Koiter

**Authors:**Wai-Kai Chen

**eBook ISBN:**9781483164151

**Imprint:**North Holland

**Published Date:**1st January 1976

**Page Count:**558

**View all volumes in this series:**North-Holland Series in Applied Mathematics and Mechanics

## Table of Contents

Chapter 1. Basic Theory

1. Introduction

2. Basic Concepts of Abstract Graphs

2.1. General Definitions

2.2. Isomorphism

2.3. Connectedness

2.4. Rank and Nullity

2.5. Degrees

3. Operations on Graphs

4. Some Important Classes of Graphs

4.1. Planar Graphs

4.2. Separable and Nonseparable Graphs

4.3. Bipartite Graphs

5. Directed Graphs

5.1. Basic Concepts

5.2. Directed-Edge Sequence

5.3. Outgoing and incoming Degrees

5.4. Strongly-Connected Directed Graphs

5.5. Some Important Classes of Directed Graphs

6. Mixed Graphs

7. Conclusions

Problems

Chapter 2. Foundations of Electrical Network Theory

1. Matrices and Directed Graphs

1.1. The Node-Edge incidence Matrix

1.2. The Circuit-Edge incidence Matrix

1.3. The Cut-Edge incidence Matrix

1.4. interrelationships Among The Matrices A, Bf, and Qf

1.5. Vector Spaces Associated with the Matrices Ba and Qa

2. The Electrical Network Problem

3. Solutions of the Electrical Network Problem

3.1. Branch-Current and Branch-Voltage Systems of Equations

3.2. Loop System of Equations

3.3. Cut System of Equations

3.4. Additional Considerations

4. invariance and Mutual Relations of Network Determinants and the Generalized Cofactors

4.1. A Brief History

4.2. Preliminary Considerations

4.3. The Loop and Cut Transformations

4.4. Network Matrices

4.5. Generalized Cofactors of the Elements of the Network Matrix

5. invariance and the incidence Functions

6. Topological Formulas for RLC Networks

6.1. Network Determinants and Trees and Cotrees

6.2. Generalized Cofactors and 2-Trees and 2-Cotrees

6.3. Topological Formulas for RLC Two-Port Networks

7. The Existence and Uniqueness of the Network Solutions

8. Conclusions

Problems

Chapter 3. Directed-Graph Solutions of Linear Algebraic Equations

1. The Associated Coates Graph

1.1. Topological Evaluation of Determinants

1.2. Topological Evaluation of Cofactors

1.3. Topological Solutions of Linear Algebraic Equations

1.4. Equivalence and Transformations

2. The Associated Mason Graph

2.1. Topological Evaluation of Determinants

2.2. Topological Evaluation of Cofactors

2.3. Topological Solutions of Linear Algebraic Equations

2.4. Equivalence and Transformations

3. The Modifications of Coates and Mason Graphs

3.1. Modifications of Coates Graphs

3.2. Modifications of Mason Graphs

4. The Generation of Subgraphs of a Directed Graph

4.1. The Generation of 1-Factors and 1-Factorial Connections

4.2. The Generation of Semifactors and k-Semifactors

5. The Eigenvalue Problem

6. The Matrix inversion

7. Conclusions

Problems

Chapter 4. Topological Analysis of Linear Systems

1. The Equicofactor Matrix

2. The Associated Directed Graph

2.1. Directed-Trees and First-Order Cofactors

2.2. Directed 2-Trees and Second-Order Cofactors

3. Equivalence and Transformations

4. The Associated Directed Graph and the Coates Graph

4.1. Directed Trees, 1-Factors, and Semifactors

4.2. Directed 2-Trees, 1-Factorial Connections, and 1-Semifactors

5. Generation of Directed Trees and Directed 2-Trees

5.1. Algebraic Formulation

5.2. Iterative Procedure

5.3. Partial Factoring

6. Direct Analysis of Electrical Networks

6.1. Open-Circuit Transfer-Impedance and Voltage-Gain Functions

6.2. Short-Circuit Transfer-Admittance and Current-Gain Functions

6.3. Open-Circuit Impedance and Short-Circuit Admittance Matrices

6.4. The Physical Significance of the Associated Directed Graph

6.5. Direct Analysis of The Associated Directed Graph

7. Conclusions

Problems

Chapter 5. Trees and Their Generation

1. The Characterizations of a Tree

2. The Codifying of a Tree-Structure

2.1. Codification by Paths

2.2. Codification by Terminal Edges

3. Decomposition into Paths

4. The Wang-Algebra Formulation

4.1. The Wang Algebra

4.2. Linear Dependence

4.3. Trees and Cotrees

4.4. Multi-Trees and Multi-Cotrees

4.5. Decomposition

5. Generation of Trees by Decomposition Without Duplications

5.1. Essential Complementary Partitions of a Set

5.2. Algorithm

5.3. Decomposition without Duplications

6. The Matrix Formulation

6.1. The Enumeration of Major Submatrices of an Arbitrary Matrix

6.2. Trees and Cotrees

6.3. Directed Trees and Directed 2-Trees

7. Elementary Transformations

8. Hamilton Circuits in Directed-Tree Graphs

9. Directed Trees and Directed Euler Lines

10. Conclusions

Problems

Chapter 6. The Realizability of Directed Graphs with Prescribed Degrees

1. Existence and Realization as a (p, s)-Digraph

1.1. Directed Graphs and Directed Bipartite Graphs

1.2. Existence

1.3. A Simple Algorithm for the Realization

1.4. Degree invariant Transformations

1.5. Realizability as a Connected (p, s)-Digraph

2. Realizability as a Symmetric (p, s)-Digraph

2.1. Existence

2.2. Realization

2.3. Realizability as Connected, Separable and Nonseparable Graphs

3. Unique Realizability of Graphs without Self-Loops

3.1. Preliminary Considerations

3.2. Unique Realizability as a Connected Graph

3.3. Unique Realizability as a Graph

4. Existence and Realization of a (p, s)-Matrix

5. Realizability as a Weighted Directed Graph

6. Conclusions

Problems

Chapter 7. State Equations of Networks

1. State Equations in Normal Form

2. Procedures for Writing the State Equations

3. The Explicit Form of the State Equation

4. An Alternative Representation of the State Equation

5. Physical interpretations of the Parameter Matrices

6. Order of Complexity

6.1 Relations between Det H(s) and Network Determinant

6.2 RLC Networks

6.3 Active Networks

7. Conclusions

Problems

Bibliography

Symbol Index

Subject Index

## Description

Applied Graph Theory: Graphs and Electrical Networks, Second Revised Edition provides a concise discussion of the fundamentals of graph and its application to the electrical network theory. The book emphasizes the mathematical precision of the concepts and principles involved. The text first covers the basic theory of graph, and then proceeds to tackling in the next three chapters the various applications of graph to electrical network theory. These chapters also discuss the foundations of electrical network theory; directed-graph solutions of linear algebraic equations; and topological analysis of linear systems. Next, the book covers trees and their generation. Chapter 6 deals with the realizability of directed graphs with prescribed degrees, while Chapter 7 talks about state equations of networks. The book will be of great use to researchers of network topology, linear systems, and circuitries.

## Details

- No. of pages:
- 558

- Language:
- English

- Copyright:
- © North Holland 1976

- Published:
- 1st January 1976

- Imprint:
- North Holland

- eBook ISBN:
- 9781483164151