## Graph Theory | by Udit Agarwal and Umesh Pal Singh | ISBN: 9788131807743. Computer Science Books. Euler and Hamiltonian Graphs. Vector Space Associated with a Graph. Detection of Planarity of a Graph. Ford and Fulkerson Algorithm

**Graph Theory**

by Udit Agarwal and Umesh Pal Singh

ISBN:9788131807743

Written in a reader-friendly style with features that enhance students’ comprehension, this concise book explains the principles and fundamental concepts of graph theory.

**Graph Theory**

Preface

**Chapter 1 – Elementary Combinatorics**

1.1: Introduction

1.2: Basic Counting Principles

1.3: The Inclusion – Exclusion Principle

1.4: Permutations (Arrangement of Objects)

1.5: Combinations

1.6: Probability

Exercises

**Chapter 2 – Discrete Numeric Functions and Generating Functions**

2.1: Discrete Numeric Functions

2.2: Manipulation of Numeric Functions

2.3: Definitions of Si a and S-i a for a Numeric Function

2.4: Generating Function

2.5: Operations on Generating Functions

2.6: Finding a Generating Function of the Fibonacci Sequence

2.7: Solution of Combinatorial Problems Using Generating Functions

Exercises

**Chapter 3 – Recurrence Relations**

3.1: Introduction

3.2: Order of Recurrence Relation

3.3: Degree of the Recurrence Relation

3.4: Solution of Linear Recurrence Relation with Constant Co-Efficients

3.5: Solution of Recurrence Relation by the Method of Generating Function

3.6: Backtracking Method

3.7: Forward Chaining Method

3.8: Summation Method

Exercises

**Chapter 4 – Graphs**

4.1: Introduction

4.2: What is a Graph ?

4.3: Basic Terminologies

4.4: Regular Graph

4.5: Complete Graph

4.6: Null Graph

4.7: Bipartite Graph

4.8: Degree Sequence

4.9: Graphs Isomorphism

4.10: Complement of a Simple Graph

4.11: Homeomorphic Graphs

4.12: Subgraphs

4.13: Operations of Graphs

4.14: Connected Graphs and Disconnected Graphs

4.15: Walk, Paths and Circuits

Exercises

**Chapter 5 – Euler and Hamiltonian Graphs**

5.1: Euler Graphs

5.2: Unicursal Graph

5.3: Constructing Eulerian Paths and Cycles

5.4: Hamiltonian Graphs

5.5: Introduction to Travelling Salesman Problem

5.6: Chinese Postman Problem

Exercises

**Chapter 6 – Trees**

6.1: Introduction

6.2: Trivial and Non-trivial Tree

6.3: Properties of Trees

6.4: Rooted Trees

6.5: Pendant Vertex

6.6: Distance and Centers in a Tree

6.7: Binary Trees

6.8: Counting Tree

6.9: Spanning Tree

6.10: Shortest Path Problems

6.11: Fundamental Circuits

Exercises

**Chapter 7 – Cut-sets and Net Work Flows**

7.1: Cut-sets

7.2: Fundamental Cut-Set

7.3: Connectivity

7.4: Separable Graph

7.5: Network-Flow Problems (Transport Network)

7.6: Ford and Fulkerson Algorithm

7.7: Cut and Its Capacity

7.8: Max-Flow Min-Cut Theorem

Exercises

**Chapter 8 – Planar and Dual Graphs**

8.1: Introduction

8.2: Kuratowski’s Graphs

8.3: Region

8.4: Euler’s Formula

8.5: Some Planar Graphs

8.6: Detection of Planarity of a Graph

8.7: Geometric Dual

8.8: Combinatorial Dual

8.9: Thickness and Crossing

Exercises

**Chapter 9 – Vector Spaces of a Graph**

9.1: Introduction

9.2: Sets with One Operation

9.3: Modular Arithmetic

9.4: Galois Fields

9.5: Vectors

9.6: Vector Space

9.7: Vector Space Associated with a Graph

9.8: Basis Vectors of a Graph

9.9: Circuit and Cut-set Subspaces

9.10: Orthogonal Vectors and Spaces

Exercises

**Chapter 10 – Matrix-Representation of Graphs**

10.1: Introduction

10.2: Incidence Matrix

10.3: Circuit Matrix

10.4: Cut-set Matrix

10.5: Relationship between AF, BF and CF

10.6: Path Matrix

10.7: Adjacency Matrix

Exercises

**Chapter 11 – Coloring of Graphs**

11.1: Graph Coloring

11.2: Chromatic Number

11.3: Chromatic Partitioning and Independent Sets

11.4: Chromatic Polynomial

11.5: Matching

11.6: Covering

11.7: Four Color Problem

11.8: The Five Color Theorem

11.9: The Six Color Theorem

Exercises

**Chapter 12 – Directed Graphs**

12.1: Introduction

12.2: Directed Graph

12.3: Types of Digraphs

12.4: Directed Paths and Connectedness

12.5: Binary Relations and Directed Graphs

12.6: Euler Directed Graphs

12.7: Trees with Directed Edges

12.8: Arborescence

12.9: Acyclic Digraphs

12.10: Fundamental Circuit in Directed Graphs

12.11: Incidence Matrix of a Digraph

12.12: Circuit Matrix of a Digraph

12.13: Adjacency Matrix of a Digraph

12.14: Orientations and Tournaments

Exercises

**Chapter 13 – Enumeration of Graphs**

13.1: Types of Enumeration

13.2: Cayley’s Theorem

13.3: Counting Unlabeled Trees

13.4: Rooted Unlabeled Trees

13.5: Free Unlabeled Trees

13.6: Permutation

13.7: Polya’s Counting Theorem

13.8: Graph Enumeration with Polya’s Theorem

Exercises

**Chapter 14 – Applications of Graph Theory**

14.1: Introduction

14.2: Graphs in Computer Science

14.3: Miscellaneous Applications

Exercises

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