Dot product graphs and domination number

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=ℤn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A=\mathbb {Z}_{n}$\end{document}, n∈ℕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n \in \mathbb {N}$\end{document}, 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 ∽\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\thicksim $\end{document} defined on R as its vertices. Also, we study the domination number of the total dot product graph of the ring R=ℤn×...×ℤn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R=\mathbb {Z}_{n}\times... \times \mathbb {Z}_{n}$\end{document}, 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.

(2020) 28:31 Page 2 of 11 distinct vertices x and y are adjacent if and only if xy ∈ I. This graph is considered to be a generalization of zero-divisor graphs of rings. In 2002, Mulay in [6] provided the idea of the zero-divisor graph determined by equivalence classes. Later on, Spiroff and Wickham in [7] denoted this graph by E (R) and compared it with (R). This graph was called the compressed graph by Anderson and LaGrange in [8] (2012). In the compressed graph, the relation on R is given by r ∼ s if and only if ann(r) = ann(s), where ann(r) = {v ∈ R | rv = 0} is the annihilator of r. This relation is an equivalence relation on R. The vertex set of the compressed graph is the set of all equivalence classes induced by ∼ except the classes [ 0] ∼ and [ 1] ∼ . The equivalence class of r is [ r] ∼ = {a ∈ R | r ∼ a} and two distinct vertices [ r] ∼ and [ s] ∼ are adjacent if and only if rs = 0. There have been other ways to associate a graph to a ring R. For surveys on the topic of zero-divisor graphs, see [9,10].
In 2015, Badawi introduced in [11] the dot product graph associated with a commutative ring R. In 2016, his student Abdulla in his master thesis [12] introduced the unit dot product graph and the equivalence dot product graph on a commutative ring with 1 = 0. We are interested here primarily in these graphs.
In 2016, Anderson and Lewis introduced the congruence-based zero-divisor graph in [13], which is a generalization of the zero-divisor graphs mentioned above. The vertices of this graph are the congruence classes of the nonzero zero-divisors of R induced by a congruence relation defined on the ring R. Two distinct vertices are adjacent if and only if their product is zero. The concept of congruence relation is used in this paper.
In 2017, Chebolu and Lockridge in [14] found all cardinal numbers occurring as the cardinality of the group of all units in a commutative ring with 1 = 0. This is very helpful to us as we want to graph the units of a ring R.
In the second section, we generalize a result of [12] concerning the unit dot product graph of a commutative ring R, where R = Z n × Z n , replacing Z n by a a commutative ring A such that U(A) is finite. In the third section, a congruence relation on the unit dot product graph is defined and some of its properties are characterized. In the last section, we discuss the domination number of some graphs.
We recall some definitions which are used in this paper. Let G be an undirected graph. Two vertices v 1 and v 2 are said to be adjacent if v 1 , v 2 are connected by an edge of G. A finite sequence of edges from a vertex v 1 of G to a vertex v 2 of G is called a path of G. We say that G is connected if there is a path between any two distinct vertices and it is totally disconnected if no two vertices in G are adjacent. For two vertices x and y in G, the distance between x and y, denoted by d(x, y), is defined to be the length of a shortest path from x to y, where d(x, x) = 0 and d(x, y) = ∞ if there is no such path. The diameter of G is diam(G) = sup{d(x, y)| x and y are vertices in G}. A cycle of length n, n ≥ 3, in G is a path of the form The girth of G, denoted by gr(G), is the length of the shortest cycle in G and gr(G) = ∞ if G contains no cycle. A graph G is said to be complete if any two distinct vertices are adjacent and the complete graph with n vertices is denoted by K n . A complete bipartite graph is a graph which may be partitioned into two disjoint nonempty vertex sets A and B such that two distinct vertices are adjacent if and only if they are in distinct vertex sets. This graph is denoted by K m,n , where |A| = m and |B| = n.
Throughout the paper, R and A denote commutative rings with 1 = 0. Its set of zero-divisors is denoted by Z(R) and Z(R) *   denote the integers, integers modulo n, and finite field with p n elements, respectively, where p is a prime number and n is a positive integer. φ(n) is the Euler phi function of a given positive integer n, which counts the positive integers up to n that are relatively prime to n.

Unit dot product graph of a commutative ring
The unit dot product graph of R was introduced in [12] , denoted by UD(R). This graph is a subgraph of the total dot product graph, denoted by TD(R), where its vertex set is all the elements of R. Some of its properties were characterized when R = A × A and A = Z n . In this section, we generalize the UD(R), as A will be a commutative ring with 1 = 0, whose multiplicative group of units is finite.
In the proof of Theorems 2 and 3, we use the order of the multiplicative group of units U(R) of R. In this context, the following theorem is helpful.

Theorem 1 (Th. 8, [14]) Let λ be a cardinal number. There exists a commutative ring R with |U(R)| = λ if and only if λ is equal to
1. An odd number of the form t i=1 (2 n i − 1) for some positive integers n 1 , ..., n t 2. An even number

An infinite cardinal number
We are interested only in commutative rings R = A × A, where A is a commutative ring with 1 = 0 and U(A) has a finite order. For instance, from [14], rings in the are examples of such a ring. Here, U(A) has an even order equal to 2 m, where m ∈ N. The units in these rings are in the form 1 + bx and −1 + bx, 0 ≤ b ≤ m − 1. If the order of U(A) is odd, then this odd number will be in the form t i=1 (2 n i −1) for some positive integers n 1 , ..., n t and the characteristic of the ring must be equal to 2.
The following two Theorems 2 and 3 characterize the graph of the rings R = R 2m × R 2m and R = A × A, respectively. [14], the units are in the form 1 + ax It is clear that every two distinct vertices in V a or in W a are not adjacent. By construction of V a and W a , every vertex in V a is adjacent to every vertex in W a . Thus, the vertices in V a ∪ W a form the graph K 2m,2m that is a complete bipartite subgraph of TD(R). By construction, UD(R) is the union of m disjoint K 2m,2m 's.
The following theorem deals with the case R = A × A, where |U(A)| is odd. In this case, the unit −1 in A (from Cauchy Theorem) must have order 1. Then, Char(A) = 2.

Congruence dot product graph of a commutative ring
In 2016, Anderson and Lewis in [13] introduced the congruence-based zero-divisor graph where ∼ is a multiplicative congruence relation on R and showed that R/ ∼ is a commutative semigroup with zero. They showed that the zero-divisor graph of R, the compressed zero-divisor graph of R, and the ideal based zero-divisor graph of R are examples of the congruence-based zero-divisor graphs of R. In this paper, we are interested in the multiplicative congruence relation ∼ on R, which is an equivalence relation on the multiplicative monoid R with the additional property that if x, y, z, w ∈ R with x ∼ y and z ∼ w, then xz ∼ yw.
The equivalence unit dot product graph of U(R) was introduced in From the definition of the congruence relation, we find that the relation defined by Abdulla is not only an equivalence relation but also a congruence relation. In fact, let c)yv and hence xw ∼ yv. We denote this congruence unit dot product graph by CUD(R), and its set of vertices is the set of all distinct congruence classes of U(R), denoted by CU(R).
In this section, we characterize the generalized case of the congruence unit dot product graph CUD(R), as we will apply the congruence relation on the unit dot product graph we introduced in the first section.
Proof For each a ∈ U(R 2m ), let V a and W a be as in the proof of Theorem 2. Then, V a , W a ∈ CU(R). Indeed, for each a ∈ U(R 2m ), there exist V a and W a ∈ CU(R) each has cardinality 2m. We conclude that each K 2m,2m of UD(R) is a K 1,1 of CUD(R). From Theorem 2 the result follows.

Example 2
In Example 1, we graphed the unit dot product graph of R 2 × R 2 , and now, we graph the congruence dot product graph of the same ring. This graph will be a complete graph of 2 vertices as R 2 is isomorphic to Z. So, we will have only two congruence classes (Fig. 2).

Theorem 5 Let R = A × A. If the order of U(A) is odd, then CUD(R) is the union of m−1
2 disjoint K 1,1 's and one K 1 . Proof For each a ∈ U(A), let V a and W a be as in the proof of Theorem 3. Then, V a , W a ∈ CU(R). Indeed, for each a ∈ U(R) and a = 1, there exist V a and W a ∈ CU(R) each of cardinality m. For a = 1, we have one congruence class V, where V = {u(a, a) | u ∈ U(A)}. We conclude that each K m,m of UD(R) is a K 1,1 of CUD(R), and each K m of UD(R) is a K 1 of CUD(R). From Theorem 3, the result follows.
Let R = Z n × Z n . We make a little change on the congruence relation defined above by taking the vertices from the whole ring R not only from U(R). Define a relation on R such that x ∼ y, where x, y ∈ R, if x = (c, c)y for some (c, c) ∈ U(R). It is clear that ∼ is an equivalence relation on R and also a congruence relation.
The congruence total dot product graph of R is defined to be the undirected graph CTD(R), and its vertices are the congruent classes of all the elements of R induced by the defined congruence relation ∼. Two distinct classes [ X] ∼ and [ Y ] ∼ are adjacent if and only if x · y = 0 ∈ Z n , where x · y denotes the normal dot product of x and y. Also, the congruence zero-divisor dot product graph, denoted by CZD(R), is defined to be an undirected graph whose vertices are the congruent classes of the nonzero zero-divisor elements in R and adjacency between distinct vertices remains as defined before.
Obviously, this congruence relation is well-defined. Indeed, let x, x , y, y ∈ R be such that y = (y 1 , y 2 ) and y = (y 1 , y 2 ) and let u, u ∈ U(R) be such that u = (c 1 , c 1 ) and u = (c 1 , c 1 ), where y 1 , y 1 , y 2 , y 2 , c 1 , c 1 ∈ Z n . Assume that x ∼ y and x ∼ y . Then, x·x = 0 if and only if (c 1 y 1 )(c 1 y 1 ) + (c 1 y 2 )(c 1 y 2 ) = 0. This happens if and only if y 1 y 1 + y 2 y 2 = 0, since c 1 c 1 is a unit in Z n . Proof If CTD(R) was connected, then ∃ x, y ∈ R such that x is adjacent to y. x·y = 0 if and only if xy = 0, leads to a contradiction with (Theorem 2.1, [11]). So, CZD(R) = ∼ (R) is connected. Since A is a field, then all the nonzero zero-divisors in R will be in two classes only, which are [ (a, 0)] ∼ and [ (0, b)] ∼ , ∀ a, b ∈ U(A) and since (a, 0) · (0, b) = 0, so it is a complete graph of two vertices.
If A = Z p and R = Z p × ... × Z p , k times and k < ∞, then the diameter and girth of CZD(R) and CTD(R) are the same as the case of TD(R) and ZD(R), which was discussed before in [11]. This reduces the number of vertices but adjacency is the same in both cases. Theorem 7 Let R = Z n × Z n for n ∈ N and n is not a prime number. Then, CTD(R) is a connected graph with diam(CTD(R)) = 3 and gr(CTD(R)) = 3.
Proof The proof is similar to that of Theorem 2.3 [11], taking into consideration that the vertices we used are in distinct classes. (b) Assume that x 2 is a zero-divisor of A, i.e., p i |x 2 in A for some p i , 1 ≤ i ≤ m. Then, v = (0, n p i ) ∈ D is adjacent to x in TD(R) (the same case takes place if x 1 is a zero-divisor of A ).
This shows that D is a dominating set of TD(R). In order to show that it is minimal, let us first remove the vertex v = (0, n p i ) from D for some i, 1 ≤ i ≤ m. We have (a, p i ) · (1, c) = a + cp i = 0 if and only if a = −cp i . So, when we remove (0, n p i ), we will find a vertex (a, p i ) for some a ∈ A which is not adjacent to any other vertices in D (as an example, take a = 1). Thus, v cannot be removed from D. The same argument is true if we remove ( n p i , 0). If we remove the unit (1, c), we will have distinct r units that are not adjacent to any other vertex in D. Thus, D is a minimal dominating set, and then, γ (TD(R)) ≤ 2m + φ(n).
We note that the upper bound of the domination number of the congruence total dot product graph of Z n × Z n 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.  (Fig. 4).
The following corollary is a generalization of Theorem 8 when R = Z n × ... × Z n , k times, k < ∞ and n is even. Proof Let x = (x 1 , ..., x k ) be a vertex in TD(R). We consider two cases: Fig. 3 Subgraph of the total dot product graph of Z 4 × Z 4