الفهرس 
Dot product graphs
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Abstract
Let A be a commutative ring with 1 ={u0338} 0 and R = A {u00D7} A. The unit dot product graph of R is de ned to be the simple graph UD(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 {u2208} A, where x · y denotes the normal dot product of x and y. In 2016, M. A. Abdulla studied this graph when A = Zn, n {u2208} N, n {u2265} 2. Inspired by this idea, we study this graph when A is a commutative ring with 1 ={u0338} 0 and a nite multiplicative group of units. We de ne the congruence unit dot product graph of R to be the simple graph CUD(R) with the congruent classes of the relation {u223C} de ned on R as its vertices. Also, we study the domination number of the total dot product graph of the ring R = Zn {u00D7}Zn, where all elements of the ring are vertices and the adjacency between two distinct vertices is the same as in UD(R). We nd an upper bound of the domi- nation number of this graph improving that found by M. A. Abdulla