Graphs from rings

This book gives an overview of research on graphs associated with commutative rings. The study of the connections between algebraic structures and certain graphs, especially finite groups and their Cayley graphs, is a classical subject which has attracted a lot of interest. More recently, attention has focused on graphs constructed from commutative rings, a field of study which has generated an extensive amount of research over the last three decades. The aim of this text is to consolidate this large body of work into a single volume, with the intention of encouraging interdisciplinary research between algebraists and graph theorists, using the tools of one subject to solve the problems of the other. The topics covered include the graphical and topological properties of zero-divisor graphs, total graphs and their transformations, and other graphs associated with rings. The book will be of interest to researchers in commutative algebra and graph theory and anyone interested in learning about the connections between these two subjects.

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 Co…