The n-Total Graph of a Commutative Ring

A Master of Science in Mathematics by Djamila Ait Elhadi entitled, “The n-Total Graph of a Commutative Ring”, submitted in November 2024. Thesis advisor is Dr. Ayman Badawi. Soft copy is available (Thesis, Completion Certificate, Approval Signatures, and AUS Archives Consent Form).

Abstract

Let n ≥ 1. This thesis introduces the n-total graph of a commutative ring R. The n-total graph of a commutative ring R, denoted by n ─ T(R), is an undirected simple graph with vertex set R, such that two vertices x; y in R are connected by an edge if xⁿ + yⁿ in Z(R). Note that if n = 1, then the 1-total graph of R is the total graph of R as in [7]. In this thesis, we study some graph properties, for example, connectedness, diameter, and girth, for the graph that we defined.