On the Unit Dot Product Graph of a Commutative Ring

On behalf of the Department of Mathematics and Statistics of the College of Arts and Sciences, you are cordially invited to a thesis defense conducted by Mohammad Abdulla of the Master of Science in Mathematics program at AUS.

Abstract:

In 2015, Ayman Badawi introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1<= n < infinity, be an integer, and R = A × A × · · · × A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R \ {(0, 0, ..., 0)}, and two distinct vertices x and y are adjacent if and only if x · y = 0 in A (where x · y denote the normal dot product of x and y). Let Z(R) denotes the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R) \ {(0, 0, ..., 0)}. Note that when n =1, then ZD(R) is just the classical zero-divisor graph of R (i.e., Beck-Anderson-Levingston graph). Let U(R) denotes the set of all units of R. Then the unit dot product graph of R is the induced subgraph UD(R) of TD(R) with vertices U(R). Let n >= 2 and A = Z_n (the integers module n). The main goal of this thesis is to study the structure of UD(R = A × A). Among many other results, the domination numbers of TD(R), ZD(R), and UD(R) are determined. We would like to point out that neither UD(R) nor its domination number were studied by Badawi. In fact, all presented results of this thesis are new.

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