The n-Total Graph of an Integral Domain

Abstract

A Master of Science in Mathematics by Myriam Abi Habib entitled, “The n-Total Graph of an Integral Domain”, submitted in May 2025. Thesis advisor is Dr. Ayman Badawi. Soft copy is available (Thesis, Completion Certificate, Approval Signatures, and AUS Archives Consent Form).Let n ≥ 1. This thesis introduces the n-total graph of an integral domain, denoted by n-TG(R) where R is an integral domain i.e. the only zero divisor of the ring R is 0. The n-total graph is an undirected graph with vertex set R. Two vertices x and y are connected by an edge if xⁿ + yⁿ Є D where D is a prime ideal or the union of prime ideals. This study will explore both the graph-theoretical and ring-theoretical properties of the n-total graph of an integral domain or a product of integral domains, such as connectedness, completeness, diameter and girth.College of Arts and SciencesDepartment of Mathematics and StatisticsMaster of Science in Mathematics (MSMTH