| Summary: | Abstract Let A be a commutative ring with 1≠0 and R=A×A. The unit dot product graph of R is defined to be the undirected graph U D(R) with the multiplicative group of units in R, denoted by U(R), as its vertex set. Two distinct vertices x and y are adjacent if and only if x·y=0∈A, where x·y denotes the normal dot product of x and y. In 2016, Abdulla studied this graph when A = ℤ n $A=\mathbb {Z}_{n}$ , n ∈ ℕ $n \in \mathbb {N}$ , n≥2. Inspired by this idea, we study this graph when A has a finite multiplicative group of units. We define the congruence unit dot product graph of R to be the undirected graph C U D(R) with the congruent classes of the relation ∽ $\thicksim $ defined on R as its vertices. Also, we study the domination number of the total dot product graph of the ring R = ℤ n × ... × ℤ n $R=\mathbb {Z}_{n}\times... \times \mathbb {Z}_{n}$ , k times and k<∞, where all elements of the ring are vertices and adjacency of two distinct vertices is the same as in U D(R). We find an upper bound of the domination number of this graph improving that found by Abdulla.
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