Some Properties of the Zero-Divisor Graphs of Idealization Ring R(+)M

e aim of this article to follow the properties of the zero-divisor graph of special idealization ring. We study the wiener index of the zero-divisors graph of some special idealization ring R(+)M and nd the clique number of the graph Γ(R(+)M) is ω (Γ(R(+) M)) |M| − 1, where R is an integral domain. We also discuss when the zero-divisors graph of some special idealization ring R(+) M are Hamiltonian graph.


Introduction
Graph theory is a very invariant tool to connect between application of mathematics and other elds. First, assume that R is a commutative ring with unity and M be an Rmodule. e ring of the idealization ring of M in R is R (+) M (r 1 , n 1 ): r 1 ∈ R, n 1 ∈ M and any two elements (r 1 , n 1 ), (r 2 , n 2 ) ∈ R(+)M which is de ned by (r 1 , n 1 )+ (r 2 , n 2 ) (r 1 + r 2 , n 1 + n 2 ) and (r 1 , n 1 )(r 2 , n 2 ) (r 1 r 2 , r 1 n 2 + r 2 n 1 ). For more details about the idealization ring, one can see in [1]. In [2,3], Anderson, Livingston and Naseer proved that the graph Γ(R) is connected with diameter at most 3. In [4] Axtell studied the zero-divisor of a commutative ring. e zero divisor graph of a commutative ring has been studied extensively by several authors [5][6][7].
Let G be a simple connected undirected graph with the vertex set V(G).
Let Γ(R) be a notation of the zero-divisor graph of a commutative ring R whose vertices are the nonzero zerodivisors of R i.e, Z * (R), with r 1 and s 1 adjacent if r 1 ≠ s 1 and r 1 s 1 0 was introduced by I. Beck in [8], who linked some algebraic properties of G with combinatorial properties of its zero-divisor graph.
Recently, Allabadi [9,10], studied the properties of the zero-divisors graph of idealization ring such as when the zero-divisors graph of idealization ring is Planar graph, divisor graph and Eulerian graph and the independence number of the zero-divisor graph of the idealization ring.
In this article we study the wiener index which is de ned the sum of all distances between vertices of the graph and is denoted by W(G).
We assume in this article the annihilator of M is ann

Wiener Index of the Graph Γ(R( + )M)
In this section, we compute the wiener index of the zerodivisor graph Γ(R(+)M).
Al-Labadi M in [10], characterized the zero-divisor graph of idealization ring when R is an integral domain. Proof. We have the following: (1) If ann(M) 0, then by [10] we have R Z 2 i.e, and d((r i , t 1 ), (r j , t 2 )) � 2 for all r i , r j ∈ ann(M), t 1 and t 2 ∈ Z 2 . Hence, the wiener index of the graph Figure 1. Proof. We have the following: (1) If ann(M) � 0, then by [10] we have R � Z 3 i.e, 1 and d((r i , t 1 ), (r j , t 2 )) � 2 for all r i , r j ∈ ann(M). Hence, the wiener index of the graph

Proof.
We have the following: (1) If ann(M) � 0, then by [10] we have In this section, we compute the wiener index of the graph [9] Al-Labadi Manal presented the zero-divisors graph of idealization ring Z N (+)Z M .
First, we compute the wiener index of the graph Hence, the wiener index of the graph Now, we characterize the sets of the zero-divisor graph for the following.
For N � p α and M � p where p is a prime number and α > 2.
where ϕ is an Euler function. □ Theorem 6. If M � p and N � p α where p is a prime number and α > 2, then the wiener index of the graph is (0,1) (0,2) (r,2) (r,1) (r,0) Proof. Let M � p and N � p α , α > 2. en the zero divisor graph of idealization ring For any vertex v i ∈ L 1 , let v j ∈ L j , 0 ≤ j ≤ α − 1. en we can define: And so on, for any vertex v i ∈ L ⌊α/2⌋ , let v j ∈ L j , 0 ≤ j ≤ α − 1. en we can define: For any vertex v i ∈ L ⌊α/2⌋ , let v j ∈ L j . en we can define: Hence from (1)- (3) and (4) the wiener index of the graph is W(Γ(Z p α (+)Z p )) � α− 1 k�0 β k . Now, we characterize the sets of the zero-divisor graph for the following.
For N � p 1 p 2 and M � p 1 where p 1 and p 2 are a prime numbers.
Proof. Let M � p 1 and N � p 1 p 2 where p 1 and. p 2 . are prime numbers. en the zero divisor graph of idealization ring Z p 1 For any vertex v i ∈ L 0 , let v j ∈ L j for all j ∈ 0, 1, 2 { }. en we can define For any vertex v i ∈ L 1 , let v j ∈ L j for all j ∈ 0, 1, 2 { }. en we can define Hence, from (5), (6) and (7) the wiener index of the graph

When is the Graph Γ(R( + )M) Hamiltonian?
In this section, we determine when the graph Γ(R(+)M) is a Hamiltonian graph.

Definition 1. A Hamiltonian cycle is a cycle that visits each vertex only once time. Definition 2. A Hamiltonian graph is a graph that contains a Hamiltonian cycle.
We beginning with the following lemma when R is an integral domain.

Lemma 1. Let R be an integral domain and M be an R− module with |M| � 2. en the graph Γ(R(+)M) is not a Hamiltonian graph.
Proof. We have the following: (1) If ann(M) � 0, then by [10], we have R � Z 2 i.e, Γ(Z 2 (+)Z 2 ) � (0, 1) { } is an empty graph which is not a Hamiltonian graph.  Proof. Let M � p 1 and N � p 1 p 2 where p 1 and p 2 are prime numbers. en the zero divisor graph of idealization ring For any vertex v i ∈ L 2 is adjacent only to any other vertex in L * 1 and any vertex in L 1 \L * 1 is adjacent only to vertex in L 0 . Hence we can not find a cycle between all vertices in the graph Γ(Z p 1 p 2 (+)Z p 1 ) that is not a Hamiltonian graph. See, Figure 3.      Proof. We have the following: (1) If ann(M) � 0, then by [10] we have R � Z 3 i.e, Γ(R(+)Z 3 ) � (0, 1), (0, 2) { } that is ω(Γ(R(+)M)) � 2.

Conclusion and Questions
In this section, we give the conclusion of the properties of the zero-divisor graph. In the future work can be ask the following questions: (1) What is the clique number for the zero-divisors graph of idealization ring Γ(R(+)M) when R is not integral domain ? (2) What is the geodetic number of the zero-divisors graph of idealization ring and the complement ?

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.