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.

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.

The concept of dominating sets introduced by Ore and Berge, is currently receiving much attention in the literature of graph theory. Several types of domination parameters have been studied by imposing several conditions on dominating sets. Ore observed that the complement of every minimal dominating set of a graph with minimum degree at least one is also a dominating set. This implies that every graph with minimum degree at least one has two disjoint dominating sets. Recently several authors initiated the study of the cardinalities of pairs of disjoint dominating sets in graphs. The inverse domination number is the minimum cardinality of a dominating set whose complement contains a minimum dominating set. Motivated by the inverse domination number, there are studies which deals about two disjoint domination number of a graph.

For the last thirty years, due to the involvement in areas such as computer science, electrical and computer engineering and operations research, graph theory has grown exponentially. Study of domination is one of the important sub-areas of graph theory and seen enormous growth due to its varied applications. The study on domination not only restricted to domination parameters, but also relates to the other parameters like independence number, covering number and others. This book deals with certain new domination parameters and their properties with other existing parameters of graphs.

Matrices play a vital role in modeling because of the rich techniques available in the domain of matrices. In this aspect, role of the inverse of a matrix is very important and is the fundamental for solution techniques. For a given matrix, the Moore-Penrose inverse is the unique matrix satisfying four fundamental matrix equations. The concept of unitary matrices for non-singular category has been extended as partial isometry to rectangular matrices, via the tool of Moore-Penrose inverses. This beginning has subsequently extended the concept of partial isometry to star-dagger matrices, which coincides with normal matrices in the case of non-singular matrices. The class of hermitian positive semi-definite matrices is a subclass of hermitian matrices, which in turn a subclass of normal matrices. The class of normal matrices includes skew-hermitian, hermitian and unitary matrices. Also another generalization of hermitian matrices is the range-hermitian matrices called the class of EP matrices.

Parallel processing and supercomputing continue to exert great influence in the development of modern science and engineering. The network of processors and interconnections play a vital role in facilitating the communication between processors in a parallel computer. Some of the popular interconnection schemes are rings, toroids and hypercubes. Their popularity stems from the commercial availability of machines with these architectures. These three families of graphs viz., rings, toroids and hypercubes share a common property of being a Cayley graph. Many important problems in networks have been modeled by Cayley graphs. One of the principal issues concerning routing problems is identification of perfect dominating sets in Cayley graphs. Circulant graphs are Cayley graphs constructed on finite cyclic groups. This book deals with domination in circulant graphs in general and some methodologies to determine dominating sets, independent dominating sets, total dominating sets and connected dominating sets in circulant graphs constructed from certain specified generating sets in particular. Domination in directed circulant graph is also dealt with.