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 is the first monograph devoted to the construction of graphs from commutative rings

Gives an overview of three decades of research in the subject

Encourages interdisciplinary work between researchers in commutative algebra and discrete mathematics

Autorentext

David F. Anderson is Professor Emeritus of Mathematics at The University of Tennessee-Knoxville, a Fellow of the American Mathematical Society, and now lives in Murfreesboro, Tennessee. He received his BS degree from Iowa State University in 1971 and PhD from the University of Chicago in 1976. His research interests are in commutative ring theory, particularly in divisibility and (non-unique) factorization in integral domains and associating graphs to commutative rings. He has written numerous research articles on these and related topics.

T. Asir is the Assistant Professor and Head at Department of Mathematics-DDE, Madurai Kamaraj University, India. His research interests are in algebraic graph theory and combinatorial commutative algebra. He has published 23 research articles in WoS-SCIE journals. His research work has been supported by various funding agencies like DST, UGC, SERB, CSIR and DSR-Saudi Arabia. He has delivered more than 50 invited talks at various conferences. Also, he has developed online mathematical content through MOOC Courses "Modern Algebra" and "Graph Theory" in SWAYAM-MHRD. Orcid-id: https://orcid.org/0000-0002-9035-6996

Ayman Badawi is a Professor at the Department of Mathematical and Statistical Sciences at the American University of Sharjah, UAE. He holds a Ph.D. in Mathematics from the University of North Texas, Texas, USA. His research interests are in the area of commutative ring theory and graphs associated to rings. Dr. Badawi is the editor in chief of the Palestine Journal of Mathematics (PJM). He has numerous publications, including book chapters, journal articles, and conference proceedings. A complete list of his works can be found on the university webpage: https://www2.aus.edu/facultybios/profile.php?faculty=abadawi and www.ayman-badawi.com

T. Tamizh Chelvam is at present a CSIR Emeritus Scientist, Government of India and formerly Senior Professor, Department of Mathematics, Manonmaniam Sundaranar University, Tamil Nadu, India. His fields of specialization are Regularity conditions in Generalized Rings and Graphs from Algebraic Structures. More specifically he has been working on Cayey graphs from finite groups, zero-divisor graphs, total graphs and generalized additive graphs from commutative rings. He has published 140 research papers and most of his papers appeared in journals like Communications in Algebra, Discrete Applied Mathematics, Journal of Algebra and Applications, Applied Mathematics Letters, Houston Journal of Mathematics, Acta Mathematica Hungarica etc. He has visited several countries like Taiwan, Singapore, UAE, Dubai and Hungary to deliver lectures in International Conferences. He is reviewer of many international journals related to Algebra and Discrete Mathematics also member of the Editorial Board of journals. The complete curriculum-vitae can be found on the webpages: http://orcid.org/0000-0001-5049-5126 and https://www.msuniv.ac.in/Academic/Department/Mathematics/Faculty

Zusammenfassung

"This book is really attractive, comprehensive, and full of valuable results for anyone working on the borderline between ring theory and graph theory." (Mehrdad Nasernejad, Mathematical Reviews, April, 2023) "This book is a very good survey of graphs defined on rings. The authors have collected results on almost all types of graphs related to rings. The book consists of ten chapters and an exhaustive bibliography of 443 items." (S. K. Nimbhorkar, zbMATH 1486.13001, 2022)

Inhalt
Introduction.- Distances in zero-divisor graphs.- Properties of zero-divisor graphs.- Genus of zero-divisor graphs.- Zero-divisor graph generalizations.- Total graphs of commutative rings.- Graphs from total graphs.- Generalized total graphs.- Other graphs associated with rings.- Bibliography.- Index.