Regular Divisor Graph of Finite Commutative Ring

Let R be a finite commutative ring with identity 1. We introduce a new graph called regular divisor graph and denoted by 𝕽 𝟃 (𝑅) . We classify the finite commutative ring to get a special graph and we are going to study some properties of this graph, clique number, chromatic number, number of cycles, connectivity and blocks.


Introduction
Let R be a finite commutative ring with identity 1, an element  ∈  is called Von Neumann regular if there exist  ∈  ℎ ℎ  = .. , a Ring R is Said to be a Von Neumann regular ring if all elements in  are Von Neumann regular [1], [10].We denote the set of Von Newman regular elements by () and set of nonzero Von Neumann regular elements by  * () = () − {0} .Taloukolaei and Sahebi introduced the Von Neumann regular graph   + () of a ring R, whose vertex set consists of elements of R and two distinct vertices  and  are adjacent if and only if  +  is a Von Neumann regular element [2].By taking advantage of their work, we have defined a new graph in this way, let  be a finite commutative ring with 1.That for all nonzero elements  and  in the ring R are adjacent if and only if  = .. or  = ..   ≠  , this graph is called by regular divisor graph and denoted by   (𝑅).with vertex set (  ()) consists of elements of  * () and edge set (  ()) = {(, ) ∶  = .. or  = .. ,  ≠  ≠ 0 } .we have used some basic concepts in ring theory from [5], [8], [10]and used some basic concepts in graph theory from [2], [3], [4], [6].Definition 1.1: [3] A graph  is finite nonempty set consist of two sets, the set of Vertices () and the set of edges ().() is a non-empty set of elements named vertices.While () is the set (which is possible empty) of unordered pairs of vertices of () called edges.The order of the graph is the number of vertices which is denoted by () , that is () = |()| , and the number of edges of  is called the Size of  and is denoted by (), that is () = |()|.Definition 1.2: [4] The degree of a vertex  of a graph  is the number of all edge's incident to v in .We denote the degree of the vertex v of  by ().The Center of a graph G is the vertex  which has greatest degree.Definition 1.3: [3] A Walk  in  is an edge, starting at  1 and ending at   such that consecutive vertices in  are adjacent.A walk in which no vertex is repeated is called a Path.A path with n vertices is denoted by   .A path that begins and ends at the same vertex is called circuit.A cycle with  ≥ 3 vertices is denoted by   .Definition 1. 4 Let  be a connected graph, the eccentricity of vertex  ∈ () , denoted by () is the distance between v and a vertex furthest from v. The diameter of  is the maximum distance between the pair of vertices, and denoted by ().While the radius of  denoted by () is the minimum distance between the pair of vertices.Definition 1.5 A complete subgraph of a graph  is called clique of  .And the maximum order of a clique of  is called clique number od  and denoted by ().The girth of graph  is the size of the smallest cycle in the graph and denoted by ().Definition 1.6: [5] The chromatic number of a graph  is denoted by () .Is the minimum number of colors needed for proper vertex coloring of  . is k-chromatic if () = .(Where k is positive integer number) Definition 1.7: Let   and   be two distinct vertices of graph  1 and  2 respectability.Two vertices   and   are identified if they replaced by anew vertex  * such that all edges incident on   and   are now incident on the new vertex  * and denoted by   ⦁  .Definition 1.8 (Double identifying) is the identifying two distinct vertices in the graphs  1 and  2 , denoted by  1 ⦁⦁ 2 , and identifying an edge between two graphs say  1 ∈  1 and  2 ∈  2 is denoted by  1 ⊶  2 .

2-Regular divisor graph of commutative ring 𝑍 𝑛
The commutative ring   ,  ≥ 1 for n equal to (prime, composite, odd, or even), is regular ring or not regular ring but in each case, it has some regular elements depending on n.
To study the regular divisor graph of the commutative ring   , which is (undirected) graph and symbolized by   (  ) , where two non-zero distinct elements in   ,  and  are adjacent as a vertex if and only if  = ..   = .. for the regular elements ,  ∈   .the regular divisor graph of commutative ring   is simple, undirected loop less graph   (  ) with vertex set (  ) and edge set (  ) = {(, ):  = ..   = .. ,  ≠  ≠ 0 ∈   }.Example 1: The ring  18 which is not regular ring but have some regular elements, the non-zero regular elements  * (  18 ) = {1,24,5,7,8,9,10,11,13,14,16,17} ,the regular divisor graph   ( 18 ) is shown in figure-2.1,which is different from any other regular divisor graphs.Gingivitis and periodontitis are two conditions listed under the umbrella term periodontal disease.Periodontal disease refers to a range of conditions that affect the supporting tissues of the teeth [1].Typically, one of the first indications of gingivitis is bleeding gums, which is a common symptom of the disorder [2].In the absence of treatment, gingivitis can progress to periodontitis, which is characterized by the loss of periodontal attachment and alveolar bone and ultimately results in tooth loss.Antibiotics can be used to treat gingivitis [3].Dentists refer to the inflammation of the gums as gingivitis.It occurs as a result of inadequate tooth cleaning, which leads to the deposition of bacterial plaque on the surface of the teeth.Therefore, effective tooth brushing is vital for achieving enough food debris clearance, as it helps to avoid the formation of plaque in the future.

Figure-2.1: Regular divisor graph 𝕽 𝟃 (𝑧 18 )
The regular divisor graph of commutative ring   ,  ≥ 1 has no well-known form (certain form) it is changed with respect on n (prime, composite, odd or even) to find the certain form of the graph we must classify the ring   with respect to the order of ring () as the following: 2.1 Regular divisor graph of the ring   for all prime number  > 3 The ring   is regular ring for all prime p, since   is a division ring and every division ring is a regular ring.The regular divisor graph of the ring   is special graph, for all non-zero element  ∈   there exist  −1 ∈   such that  and  −1 are adjacent.Theorem 2.1.1The regular divisor graph of the ring   is bipartite graph and where  2 is a complete graph of order two.Proof: Since the ring   is division ring then   is regular, it means for all  ∈    is a regular element.Then the vertex set V(  ) =   − {0} , two elements 1 and The vertices ai in  1 and     2 are adjacent together as follows.
In  1 and  2 , the unit elements are adjacent together each  with   −1 and   with   −1 , but two elements 1 and 2 − 1 in  1 are their own inverse and they are adjacent with p+1 and p-1 in V2 respectively rather than the adjacency with vertex p as follow: In another hand two vertices p-1 and p+1 in  2 are two end vertices, they are of degree one.We proved that the vertex p has a maximum degree in the regular divisor graph, then  is center of the graph   ( 2 ).Remark: To describe the regular divisor graph   (  ) we need to define a new operation in graph theory we called it half join and denoted by ⨭  means the join operation between two graphs when we take half number of vertices in the second graph.Since  + 1 is one of the idempotent elements in the ring, then ( + 1) 2 =  + 1 up to the regularity of the ( + 1) its adjacent with only one vertex 1, and the other  − 1 is regular with respect to (2 − 1) also we exclude the selfregularity by the same reason.

(2𝑝 + 1) = (𝑝 + 1). (2𝑝 + 1) (also
It is worth mentioning in both cases for all  ∈  3 there is two elements in  2 such that b is adjacent with them.The distance between p with   equal to 1 or 3 for all   ∈  2 , the distance between 2p with   equal to 1 or 3 for all   ∈  2 , the distance between p with   equal to 2 for all   ∈  3 , the distance between 2p with   equal to 2 for all   ∈  3 ,the distance between   equal to 1 or 2 or 3 or 4 for all   ∈  2 , the distance between   equal to 1 or 4 for all   ∈  3 , the distance between   with   equal to 1 or 2 or 3 for all   ∈

Proposition 2 . 3 . 6 :
The regular divisor graph   ( 2 ) contains ( In the fact that we have always two end vertices  + 1 and  − 1 adjacent with 1 and 2 − 1 respectability and they are the only two vertices of degree one.All the other vertices in  1 − {} and  2 are adjacent together as follow to make the cycle  4 (  ,   −1 ) , (  ,   ~), (  ,   ),and (  −1 ,   ~)Since we have  − 1 vertices in each partite set we exclude two vertices in each partite sets then we have (

Corollary 2 . 3 . 9 :Proposition 2 . 3 . 10 : 4 Theorem 2 . 3 . 11 :
The clique number equal to girth in the graph   ( 2 ). (  ( 2 )) = ℊ (  ( 2 )) = 3 Proof: It is clear that the shortest cycle in the graph   ( 2 ) is  3 and length of this cycle is three then girth of the graph is equal to 3 and clique number=3.The dimeter of regular divisor graph   ( 2 ) ,  (  ( 2 )) = 4 .Proof: In the general form of the graph   ( 2 ) that it is shown in the figure-2.4itis clear that the distance between p with   for all   ∈  1 − {} is equal to 1, the distance between p with   for all   ∈  2 is equal to 2, the distance between   for all   ∈  1 − {} is equal to 1 or 2, the distance between   with   is equal to 1 or 2 or 3 for all   ∈  1 − {} and   ∈  2 , the distance between   is equal to 1 or 4 for all   ∈  2 , So, the maximum distance in the graph   ( 2 ) is equal to 4 , Then  (  ( 2 )) = Chromatic number  (  ( 2 )) = 3 .Proof: The vertex  is adjacent with all vertices in  1 − {} and for all   ∈  1 − {} there exists   −1 ∈  1 − {} such that   is adjacent with   −1 , then they must have different color, the vertices in  2 they are adjacent together and adjacent with some vertices in  1 by respect to the regularity  + 1,  − 1 are adjacent with 1,2 − 1 respectability, then if p and  + 1,  − 1 are red all   ,   ~ with 1 and 2 − 1 take another color say blue and   −1 with   take another color, so we use three different colors to coloring all vertices in the graph   ( 2 ) As shown in figure-2.5-.Then  (  ( 2 )) = 3 .

Figure 2 . 5 :
Figure 2.5: chromatic number for the general form in the regular divisor graph   ( 2 ).Definition 2.3.12:Butterfly graph  1, 3 is a graph obtained from 2 path  2 and n cycle  3 identifying in one vertex r called a root as shown in figure-2.6-.

Tikrit Journal of Pure Science (2023) 28 (5):158-175 Doi
The regular divisor graph of the ring  2 is planner graph and the smallest cycle in   ( 2 ) is  3 obtained from the adjacency between the vertices ,   , : https://doi.org/10.25130/tjps.v28i5.1587164 Corollary 2.3.7:The regular divisor graph   ( 2 ) for p>2, is planner graph.Proof: By theorem 2.3.4 clearly has no crossing number in the regular divisor graph   ( 2 ) then it is planner graph.Proof: −1 then complete subgraph is  3 .And the order of  3 is equal to 3.