Non-Braid Graphs of Ring

ABSTRACT


A. INTRODUCTION
Some different areas in mathematics can be combined due to obtaining some interesting tools. A new interdisciplinary branch in mathematics such as algebraic graphs recently has powerful progress, in which involves graph theory and some algebraic structures. Associating a graph with an algebraic structure is a research subject that aims at exposing the relationship between algebra and graph theory and at advancing applications of one to the other (Maimani et al., 2011). There are plenty of results in associating a graph with an algebraic structure, some of them are in (Ghalandarzadeh & Rad, 2011), (Ma et al., 2014), and (Taloukolaei & Sahebi, 2018) which results in associating a graph with the module, group, and ring, respectively.
Among many results in associating a graph with the group, we mention for example Abdolahi et al. (2006), who discuss the non-commuting graph of a group. They explore how the graph-theoretical properties of the non-commuting graph of a group can affect the group-theoretical properties of . Some important results are the number of edges of the noncommuting graph shows the association of a group to the graph, the number of edges of the non-commuting graph and commutativity degree are in the opposite proportion (Abdollahi et al., 2006). Furthermore, Tolue et al. (2014) introduce a more general situation by defining the -non-commuting graph of finite groups and investigating its planarity and regularity, its clique number, and dominating number.
Some authors also pay attention to the observation of graphs and rings. For a ring , Gupta (2013) defined a simple undirected graph Γ 2 ( ) with all the non-zero elements of as vertices, and two vertices , are adjacent if and only if either = 0 or = 0 or + is a zero-divisor (including 0). He considers the connectedness and completeness of Γ 2 (ℤ n ). Omidi and Vatandoost (2011) introduce the commuting graph of the ring R. Let R be a noncommutative ring and let Z(R) denote the center of R. The commuting graph of R denoted by Γ(R), is a graph with vertex set R\Z(R) and two vertices a and b are adjacent if ab = ba (Omidi & Vatandoost, 2011). The complement of commuting graph of the ring R is considered by Erfanian et al. (2015). They introduce the non-commuting graph of ring R and study various graph-theoretical properties of this graph. Let be the non-commutative ring and ( ) is the centre of . The non-commuting graph of ring , denoted by Γ R , is a simple graph whose vertex set is \ ( ) and two distinct vertices and are connected by one edge if and only if ≠ (Erfanian et al., 2015). Some further results regarding the non-commuting graph of a ring are conducted by Dutta and Basnet (2017a). They prove that Γ R is not isomorphic to certain graphs of any finite non-commutative ring . Moreover, some connections between Γ R and the commuting probability of are also obtained. It is shown that the non-commuting graphs of two ℤ-isoclinic rings are isomorphic if the centres of the rings have the same order (Dutta & Basnet, 2017a). Recall that two rings 1 and 2 are said to be ℤ-isoclinic if there exist additive group isomorphisms : 1 ( 1 ) → 2 ( 2 ) and : [ 1 , 1 ] → [ 2 , 2 ] such that ([ , ]) = [ ′, ′] whenever ( + ( 1 )) = ′ + ( 2 ) and ( + ( 1 )) = ′ + ( 2 ) (Dutta et al., 2015). Dutta and Basnet (2017b) introduce a generalization of the non-commuting graph for the ring, namely the relative non-commuting graph of a finite ring. Let be a subring of a finite ring and ( ) = { ∈ ∶ = ∀ ∈ }. The relative non-commuting graph of the subring in , denoted by Γ , , is a simple undirected graph whose vertex set is \ ( ) and two distinct vertices , are adjacent if and only if or ∈ and ≠ (Dutta & Basnet, 2017b). They discuss some properties of Γ , , determine the diameter, girth, some dominating sets, and chromatic index for Γ , . Another generalization of non-commuting graph for ring is introduced by Nath et al. (2021), namely the -non-commuting graph of a finite ring , for some in . This graph, which is denoted by Γ , is a simple undirected graph whose vertex set is and two vertices and are adjacent if and only if [x, y] ≠ r and [x, y] ≠ −r (Nath et al., 2021). Nath et al. prove that Γ is neither a regular graph nor a lollipop graph if is noncommutative. Moreover, they characterize the finite non-commutative ring such that Γ is a tree or a star graph.
For the detailed explanation of some notions in graph theory, we refer to (Chartrand et al., 2015) and (Wilson, 2010). Allcock (2002) introduced the notion of the braid. Two group elements and braid, if they satisfy = (Allcock, 2002). In other words, two group elements and non-braid, if they satisfy ≠ . So far, there is no graph representing the non-braid property of two elements in ring . Hence, in this paper, we will define a graph that represents that property.
By adopting the concept of defining the non-commuting graph of ring , we define a nonbraid graph of the ring based on the non-braid property of two elements in ring under multiplication. We introduce the definition of a non-braid graph of the ring as follows. Let be a ring and ( ) = { ∈ |∀ ∈ , = }. We called ( ) as the braider of . The non-braid graph of , denoted by Υ , is a simple graph whose vertex set is \ ( ). Two distinct vertices and are connected by an edge if and only if ≠ . The vertices and that satisfy these condition is denoted by ∼ .
Since the non-commuting graph of ring cannot be constructed for the commutative ring, we are interested to investigate the existence of a non-braid graph of a commutative ring, Example of a commutative ring is ring ℤ (Malik et al., 1997). Some authors already associated a graph with ring ℤ , as in (Chelvam & Asir, 2011), (Patra & Kalita, 2014), (Pirzada et al., 2020), and (Aditya & Muchtadi-Alamsyah, 2021). In this paper, we discuss the non-braid graph of ring ℤ . It is important to know the kind of non-braid graph based on of ℤ . Hence as the purpose of the study we observe the relationships of and the completeness and connectedness of the graph. We show sufficient condition of such that the non-braid graph of ℤ is complete and prove that all Υ ℤ is connected.

B. RESULT AND DISCUSSION
Gupta (2013) defined a simple undirected graph Γ 2 ( ) and determined the connectedness of Γ 2 (ℤ ). In this paper, we conduct a research in connectedness and completeness of nonbraid graph of ring ℤ . The notion of non-braid graph is motivated by commuting graph from Omidi and Vatandoost (2011) and the non-commuting graph from Erfanian et al. (2015).

Definition of Non-braid Graph
In this section, we define a non-braid graph of ring . Definition 2.1. Let be a ring and ( ) = { ∈ | ∀ ∈ , = }. A non-braid graph of ring , denoted by Υ , is a simple graph whose vertex set is \ ( ). Two distinct vertices and are connected by an edge if and only if ≠ . Furthermore, and that satisfy this condition is denoted by ~.
For a special case, we discuss the non-braid graph of ring ℤ which is denoted by Υ ℤ . We start by investigating the existence of a non-braid graph of ring ℤ . We take a note for = 1, (Υ ℤ 1 ) cannot be empty, meaning that a non-braid graph of ℤ 1 cannot be formed. We then take a note for = 2, that is However, (Υ ℤ 2 ) cannot be empty, which means that a non-braid graph of ℤ 2 cannot be constructed.

Completeness of The Non-braid Graph ℤ
We give some results regarding the completeness of the non-braid graph Υ ℤ .
Theorem 2.2. If is prime, then Υ ℤ is a complete graph.
Proof. Let , ∈ ℤ ∖ (ℤ ) where ≠ . We will show that ∼ . Assume that ≁ . It means = . Since ℤ is a field, the cancellation properties hold and we get = . This contradicts the hypothesis. Hence, ∼ . ∎ The following is an example of a non-braid graph of ring ℤ 7 which is a complete graph.

Example 2.3.
For every x ∈ ℤ 7 , we get x 0 x = 0 x 0, so 0 ∈ B(ℤ 7 ). Then, the result of x y x and y x y for every x, y ∈ ℤ 7 ∖ {0} where x ≠ y is shown in Table 3. Based on Table 3, we get that all vertices are adjacent to others. So, we get a non-braid graph Υ ℤ 7 illustrated by Figure 3.

Braider of ℤ
Regarding of connection of non-braid graph Υ ℤ we start to find out the braider of ℤ . To begin with, we study a useful property of ring ℤ for finding out the braider of ℤ . Lemma 2.3. Let ℤ 2 be a ring where ∈ ℕ and m > 1. The number is odd if and only if is an idempotent element.

Case 2.
If is an idempotent element, then we claim that there is − 1 ∈ ℤ such that − 1 − 1 ≠ − 1 . Assume that − 1 − 1 = − 1 . Since is an idempotent element, we have Subsequently, since ≠ 2 where is odd, there are two possibilities for the value of , namely, is odd or = 2 where is even and ≠ 0. It is obvious that for odd , there is no that satisfies 2 = 0, which is a contradiction. Next, for = 2 where is even and ≠ 0. It is clear that which satisfies 2 = 0 is only = . Since = is an idempotent element, by Lemma 2.3, we get that is odd, which is a contradiction.

The Connectedness of The Non-braid Graph ℤ
Now we study the adjacency properties of two vertices that will be useful for investigating the connection of non-braid graph Υ ℤ . We will find out which vertices are always adjacent to each other or not. We find that the adjacency of a vertex depends on whether it is an idempotent element or not. In the next two following lemma we will give the adjacency of idempotent element. We start by showing the necessary and sufficient condition of adjacency for any vertex with vertex 1.
Lemma 2.5. Let ∈ ℤ ∖ (ℤ ). Vertex is adjacent to vertex 1 if and only if is not an idempotent element.
(⇐) Suppose is not an idempotent element. Note that 1 1 = and 1 = 2 . Since is not an idempotent element, then 2 ≠ , which implies 1 1 ≠ 1 . Hence, vertex is adjacent to vertex 1. ∎ By Lemma 2.5, we can conclude that the vertex which is not an idempotent element is adjacent to 1. Next, we give the result of the adjacency property for the vertex , which is an idempotent element.
Proof. Suppose is an idempotent element. We will show that vertex is adjacent to vertex − 1. For ≠ 2 where is odd, by the proof of Lemma 2.4, point 1, case 2, we have − 1 ≠ − 1 − 1. Hence, vertex is adjacent to vertex − 1. Next, for = 2 where is odd. Assume that − 1 = − 1 − 1. Since is an idempotent element, then satisfies 2 = 0 by equation 1. It is clear that which satisfies 2 = 0 is only = . By Lemma 2.4, point 2, we have = ∈ (ℤ ), which is a contradiction. Hence, for = 2 where is odd, we have − 1 ≠ − 1 − 1 and consequently, vertex is adjacent to vertex − 1. Therefore, it is proven that vertex is adjacent to vertex − 1. ∎ Adjacency of any vertices depends on not only an idempotent element, but also a unit element. In the following lemma, we show the adjacency properties of two vertices, which are unit elements.
Proof. Suppose and are unit elements where ≠ . Assume that vertex is not adjacent to vertex . It means that = . Note that u y u = y u y u u y = u y y u −1 u u y y −1 = u −1 u y y y −1 u = y This is a contradiction. Hence, vertex u is adjacent to vertex y. ∎ By using Lemma 2.5, 2.6, and 2.7, we give the result of the connection of non-braid graph Υ ℤ for any .
Theorem 2.8. If ≥ 3, then Υ ℤ n is a connected graph. Proof. Let , ∈ ℤ ∖ (ℤ ) where ≠ . We will show that there is a path from vertex to vertex . We consider two cases. Case 1. If and are unit elements, then by Lemma 2.7, vertex is adjacent to vertex . Case 2. For or which is a zero-divisor. Without loss of generality, suppose is a zero-divisor. Since every element of ℤ is either an idempotent element or not, then zero-divisor is either an idempotent element or not. By Lemma 2.5, 2.6, 2.7, we have the following result. If is not an idempotent element, then a. for a zero-divisor which is not an idempotent element or which is a unit element where ≠ 1, we can construct a path − 1 − , …(2) b. for = 1, we have that vertex is adjacent to vertex , c. for a zero-divisor y which is an idempotent element, we can construct a path If is an idempotent element, then a. for a zero-divisor which is not an idempotent element, we can construct a path − − 1 − 1 − , …(4) b. for a zero-divisor which is an idempotent element or which is a unit element where ≠ − 1, we can construct a path − − 1 − , …(5) c. for = − 1, we have that vertex is adjacent to vertex .
By cases 1 and 2, it is proven that there is a path from vertex to vertex . Hence, Υ ℤ is a connected graph. ∎

C. CONCLUSION AND SUGGESTIONS
If is a prime number, the non-braid graph Υ ℤ is a complete graph. For all ≥ 3 the nonbraid graph Υ ℤ is a connected graph.