On The Unit Dot Product Graph Of A Commutative Ring.

Abstract

In 2015, Ayman Badawi (Badawi, 2015) introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 n < 1 be an integer, and R =A x A x ... x A (n times). We recall from (Badawi, 2015) that total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R \ {(0, 0, ..., 0)}, and two distinct vertices x and y are adjacent if and only if xy = 0 [is an element of] A (where xy 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) = Z(R)* \ {(0, 0, ..., 0)}. 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 = Zn. The main goal of this thesis is to study the structure of UD(R = A x A).