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Table of Contents
Intro
Preface
Notation
Contents
Chapter 1 Introduction
1.1 Basics in Algebra
1.2 Basics in Graph Theory
Chapter 2 Distances in Zero-divisor Graphs
2.1 Zero-divisor Graphs
2.2 Basic Properties of Zero-divisor Graphs
2.3 Girth of Zero-divisor Graphs
2.3.1 Rings of Polynomials and Power Series
2.3.2 Idealization
2.3.3 Rings with Linearly Ordered Primes
2.4 Diameter of Zero-divisor Graphs
2.4.1 Rings of Polynomials and Power Series
2.4.2 Idealization
2.4.3 Rings with Linearly Ordered Primes
2.5 Radius and Center of Zero-divisor Graphs
2.6 Rings without Identity
Chapter 3 Properties of Zero-divisor Graphs
3.1 Coloring of Zero-divisor Graphs
3.1.1 Vertex Coloring
3.1.2 Clique Number
3.1.3 Edge Coloring
3.2 Connectivity of Zero-divisor Graphs
3.3 Isomorphisms and Automorphisms
3.4 Bipartite Zero-divisor Graphs
3.5 Zero-divisor Graphs of Gaussian Integers
3.6 Cycles in Zero-divisor Graphs
3.6.1 Trees and Complements
3.6.2 Eulerian and Hamiltonian Graphs
Chapter 4 Genus of Zero-divisor Graphs
4.1 Genus of Graphs
4.2 Planar Zero-divisor Graphs
4.3 Zero-divisor Graphs with Genus One
5.9 Compressed Zero-divisor Graphs
5.9.1 Properties of Compressed Zero-divisor Graphs
5.9.2 Girth of Compressed Zero-divisor Graphs
5.10 Extended Zero-divisor Graphs
Chapter 6 Zero-divisor Graph Generalizations
6.1 Noncommutative Rings
6.1.1 Directed Zero-divisor Graphs of Finite Rings
6.2 Commutative Semigroups
6.3 Commutative Semirings
6.4 Modules
6.4.1 Torsion Graphs of Modules
6.4.2 Zero-divisor Graphs of Modules
6.4.3 Annihilator Ideal-based Graphs of Modules
6.5 Domination in Directed Zero-divisor Graphs
Chapter 7 Total Graphs of Commutative Rings
7.1 Definition of Total Graph
7.2 Basic Properties of Total Graphs
7.3 Polynomial Rings and Idealization
7.4 Properties of Total Graphs
7.4.1 Eulerian and Hamiltonian Graphs
7.4.2 Colorings
7.4.3 Connectivity
7.4.4 Domination
7.4.5 Intersection Graphs of Dominating Sets
7.5 Genus of Total Graphs
Chapter 8 Graphs from Total Graphs
8.1 Total Graphs without the Zero Element
8.1.1 Basic Properties
8.1.2 Zero-divisor Paths and Regular Paths
8.2 The Regular Graph
8.3 Complements of Total Graphs
8.4 Generalized Complements of Total Graphs
Preface
Notation
Contents
Chapter 1 Introduction
1.1 Basics in Algebra
1.2 Basics in Graph Theory
Chapter 2 Distances in Zero-divisor Graphs
2.1 Zero-divisor Graphs
2.2 Basic Properties of Zero-divisor Graphs
2.3 Girth of Zero-divisor Graphs
2.3.1 Rings of Polynomials and Power Series
2.3.2 Idealization
2.3.3 Rings with Linearly Ordered Primes
2.4 Diameter of Zero-divisor Graphs
2.4.1 Rings of Polynomials and Power Series
2.4.2 Idealization
2.4.3 Rings with Linearly Ordered Primes
2.5 Radius and Center of Zero-divisor Graphs
2.6 Rings without Identity
Chapter 3 Properties of Zero-divisor Graphs
3.1 Coloring of Zero-divisor Graphs
3.1.1 Vertex Coloring
3.1.2 Clique Number
3.1.3 Edge Coloring
3.2 Connectivity of Zero-divisor Graphs
3.3 Isomorphisms and Automorphisms
3.4 Bipartite Zero-divisor Graphs
3.5 Zero-divisor Graphs of Gaussian Integers
3.6 Cycles in Zero-divisor Graphs
3.6.1 Trees and Complements
3.6.2 Eulerian and Hamiltonian Graphs
Chapter 4 Genus of Zero-divisor Graphs
4.1 Genus of Graphs
4.2 Planar Zero-divisor Graphs
4.3 Zero-divisor Graphs with Genus One
5.9 Compressed Zero-divisor Graphs
5.9.1 Properties of Compressed Zero-divisor Graphs
5.9.2 Girth of Compressed Zero-divisor Graphs
5.10 Extended Zero-divisor Graphs
Chapter 6 Zero-divisor Graph Generalizations
6.1 Noncommutative Rings
6.1.1 Directed Zero-divisor Graphs of Finite Rings
6.2 Commutative Semigroups
6.3 Commutative Semirings
6.4 Modules
6.4.1 Torsion Graphs of Modules
6.4.2 Zero-divisor Graphs of Modules
6.4.3 Annihilator Ideal-based Graphs of Modules
6.5 Domination in Directed Zero-divisor Graphs
Chapter 7 Total Graphs of Commutative Rings
7.1 Definition of Total Graph
7.2 Basic Properties of Total Graphs
7.3 Polynomial Rings and Idealization
7.4 Properties of Total Graphs
7.4.1 Eulerian and Hamiltonian Graphs
7.4.2 Colorings
7.4.3 Connectivity
7.4.4 Domination
7.4.5 Intersection Graphs of Dominating Sets
7.5 Genus of Total Graphs
Chapter 8 Graphs from Total Graphs
8.1 Total Graphs without the Zero Element
8.1.1 Basic Properties
8.1.2 Zero-divisor Paths and Regular Paths
8.2 The Regular Graph
8.3 Complements of Total Graphs
8.4 Generalized Complements of Total Graphs