دانلود کتاب Graphs from Rings

فهرست مطالب :

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
4.4 Zero-divisor Graphs with Genus Two
4.5 Zero-divisor Graphs with Large Genus
4.6 Crosscap of Zero-divisor Graphs
4.7 Embedding of Line Graphs
Chapter 5 Generalized Zero-divisor Graphs
5.1 Ideal-based Zero-divisor Graphs
5.2 Properties of ΓI(R)
5.2.1 Clique Number of Ideal-based Zero-divisor Graphs
5.3 Genus of Ideal-based Zero-divisor Graphs
5.4 Primal Ideal-based Zero-divisor Graphs
5.5 Annihilating-ideal Graphs
5.6 Special Annihilating-ideal Graphs
5.7 Coloring of Annihilating-ideal Graphs
5.8 Genus of Annihilating-ideal Graphs
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
8.5 Line Graphs of Total Graphs
Chapter 9 Generalized Total Graphs
9.1 Commutative Ring Generalizations
9.1.1 Ideal-based Generalized Total Graphs
9.1.2 Total Graphs of Multiplicatively Closed Sets
9.1.3 Total Graphs of Multiplicative-prime Sets
9.2 Total Graphs of Modules
9.2.1 Total Torsion Element Graphs
9.2.2 Total Graphs of Modules
9.3 Commutative Semirings
9.3.1 Total Graphs of Semirings
9.3.2 Identity Summand Graphs
Chapter 10 Other Graphs Associated with Rings
10.1 Graphs of Commutative Rings
10.1.1 Cayley Graphs
10.1.2 Co-maximal and Unit Graphs
10.1.3 Cozero-divisor Graphs
10.1.4 Jacobson and Intersection Graphs
10.2 Annihilator Graphs
10.2.1 Equal Annihilator and Zero-divisor Graphs
10.3 Compressed Annihilator Graphs
10.4 ΓI(R), ΓI(R), and AGI(R)
10.5 Congruence-based Graphs
10.6 Element Graphs
10.7 Dot Product Graphs
10.7.1 Subgraphs of Dot Product Graphs
10.7.2 Equivalence Dot Product Graphs
10.8 Trace Graph of Matrices
10.8.1 Basic Properties of Trace Graphs
10.8.2 Planarity and Thickness of Trace Graphs
10.8.3 Domination in Trace Graphs of Matrices
10.8.4 Ideal-based Trace Graphs of Matrices
10.9 Graphs of Noncommutative Rings
10.9.1 Commuting and Non-commuting Graphs
10.9.2 Nilpotent and Idempotent Graphs
10.10 Graphs of Ideals
10.11 Divisor Graphs
10.12 Subgroup Graphs of a Group
Bibliography
Index