In 2016, Anderson and Lewis in [13] introduced the congruence-based zero-divisor graph \(\Gamma _{\thicksim }(R)=\Gamma (R/\thicksim)\), where \(\thicksim \) is a multiplicative congruence relation on R and showed that \(R/\thicksim \) 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 \(\thicksim \) on R, which is an equivalence relation on the multiplicative monoid R with the additional property that if x,y,z,w∈R with \(x\thicksim y\) and \(z\thicksim w\), then \(xz\thicksim yw\).
The equivalence unit dot product graph of U(R) was introduced in [12], where R=A×A and \(A=\mathbb {Z}_{n}\). The equivalence relation \(\thicksim \) on U(R) was defined such that \(x\thicksim y\), where x,y∈U(R), if x=(c,c)y for some (c,c)∈U(R). Let EU(R) be the set of all distinct equivalence classes of U(R). If X∈EU(R), then ∃ a∈U(A) such that \(X=[(1,a)]_{\thicksim }=\{u(1,a)\,|\,u\in U(A)\}\). Thus, the equivalence unit dot product graph of U(R) is the (undirected) graph EUD(R) with vertices EU(R). 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.
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 \(x \thicksim y\) and \(w \thicksim v\). So, x=(c1,c1)y and w=(c2,c2)v for some (c1,c1),(c2,c2)∈U(R). Then, xw=(c1,c1)y(c2,c2)v=(c1,c1)(c2,c2)yv=(c,c)yv and hence \(xw \thicksim yv\). 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.
Theorem 4
Let R=R2m×R2m. Then, CUD(R) is the union of m disjoint K1,1’s.
Proof
For each a∈U(R2m), let Va and Wa be as in the proof of Theorem 2. Then, Va,Wa∈CU(R). Indeed, for each a∈U(R2m), there exist Va and Wa∈CU(R) each has cardinality 2m. We conclude that each K2m,2m of UD(R) is a K1,1 of CUD(R). From Theorem 2 the result follows. □
Example 2
In Example 1, we graphed the unit dot product graph of R2×R2, and now, we graph the congruence dot product graph of the same ring. This graph will be a complete graph of 2 vertices as R2 is isomorphic to \(\mathbb {Z}\). So, we will have only two congruence classes \([(1,1)]_{\thicksim }=\{(1,1),(-1,-1)\}\) and \([(1,-1)]_{\thicksim }=\{(1,-1),(-1,1)\}\) (Fig. 2).
Theorem 5
Let R=A×A. If the order of U(A) is odd, then CUD(R) is the union of \(\frac {m-1}{2}\) disjoint K1,1’s and one K1.
Proof
For each a∈U(A), let Va and Wa be as in the proof of Theorem 3. Then, Va,Wa∈CU(R). Indeed, for each a∈U(R) and a≠1, there exist Va and Wa∈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 Km,m of UD(R) is a K1,1 of CUD(R), and each Km of UD(R) is a K1 of CUD(R). From Theorem 3, the result follows. □
Let \(R=\mathbb {Z}_{n}\times \mathbb {Z}_{n}\). 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 \(x \thicksim y\), where x,y∈R, if x=(c,c)y for some (c,c)∈U(R). It is clear that \(\thicksim \) 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 \(\thicksim \). Two distinct classes \([X]_{\thicksim }\) and \([Y]_{\thicksim }\) are adjacent if and only if \(x\cdot y= 0\in \mathbb {Z}_{n}\), 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=(y1,y2) and y′=(y1′,y2′) and let u,u′∈U(R) be such that u=(c1,c1) and u′=(c1′,c1′), where \(y_{1}, y_{1}', y_{2}, y_{2}', c_{1}, c_ 1' \in \mathbb {Z}_{n}\). Assume that \(x\thicksim y\) and \(x'\thicksim y'\). Then, x·x′=0 if and only if (c1y1)(c1′y1′)+(c1y2)(c1′y2′)=0. This happens if and only if y1y1′+y2y2′=0, since c1c1′ is a unit in \(\mathbb {Z}_{n}\).
Theorem 6
Let \(A=\mathbb {Z}_{p}\), where p is a prime number and R=A×A. Then, CTD(R) is disconnected and \(CZD(R)=\Gamma _{\thicksim }(R)\) is a complete graph of 2 vertices.
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, \(CZD(R)=\Gamma _{\thicksim }(R)\) is connected. Since A is a field, then all the nonzero zero-divisors in R will be in two classes only, which are \([(a,0)]_{\thicksim }\) and \([(0,b)]_{\thicksim }\), ∀ a,b∈U(A) and since (a,0)·(0,b)=0, so it is a complete graph of two vertices. □
If \(A=\mathbb {Z}_{p}\) and \(R=\mathbb {Z}_{p}\times... \times \mathbb {Z}_{p}\), 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.
Example 3
If \(A=\mathbb {Z}\), then R=\(\mathbb {Z}\times \mathbb {Z}\). Here, the only units in the form (c,c) are (1,1) and (-1,-1) so the classes of the zero-divisors will be in the form \([(a,0)]_{\thicksim }=\{(a,0),(-a,0)\}\) and \([(0,a)]_{\thicksim }=\{(0,a),(0,-a)\}\), ∀ a∈U(A). For two distinct vertices (a,0)·(b,0)≠0, because ab≠0. Then, there will be an edge only between classes in the form \([(a,0)]_{\thicksim }\) and \([(0,b)]_{\thicksim }\), which means diam(CZD(R))=2 and gr(CZD(R))=4.
Theorem 7
Let \(R=\mathbb {Z}_{n}\times \mathbb {Z}_{n}\) for \(n\in \mathbb {N}\) 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. □