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 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∈A, where x·y denotes the normal dot product of x and y. In 2016, Abdulla studied this graph when A=mathbb {Z}_{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 CUD(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=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 UD(R). We find an upper bound of the domination number of this graph improving that found by Abdulla.
Highlights
Introduction and preliminariesThe idea of a zero-divisor graph of a commutative ring R was introduced by Beck in [1] (1988)
We note that the upper bound of the domination number of the congruence total dot product graph of Zn × Zn is the same as the previous result of the total dot product graph, taking into consideration that the vertices we used are in distinct classes
We note that the upper bound of the domination number of the congruence total dot product graph of Zn × ... × Zn is the same as the previous result of the total dot product graph, taking into account that the vertices we used are in distinct classes
Summary
In both graphs, every unit (1, c) is adjacent to r units in the form u(1, c) for all u ∈ U(A), case takes place if x1 is a zero-divisor of A ). Every unit (1, c) is adjacent to r units in the form u(1, c) for all u ∈ U(A), case takes place if x1 is a zero-divisor of A ) This shows that D is a dominating set of TD(R). We note that the upper bound of the domination number of the congruence total dot product graph of Zn × Zn is the same as the previous result of the total dot product graph, taking into consideration that the vertices we used are in distinct classes.
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