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

Authors: Manal Al-Labadi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The 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 find the clique number of the graph Γ(R(+)M) isω (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.


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## Body

## 1. Introduction

Graph theory is a very invariant tool to connect between application of mathematics and other fields. First, assume thatR is a commutative ring with unity and M be an R- module. The ring of the idealization ring of M in R is R (+) M=r1,n1:r1∈R,n1∈M and any two elements r1,n1, r2,n2∈R+M which is defined by r1,n1+r2,n2=r1+r2,n1+n2 and r1,n1r2,n2=r1r2,r1n2+r2n1. For more details about the idealization ring, one can see in [1]. In [2, 3], Anderson, Livingston and Naseer proved that the graph ΓR is connected with diameter at most 3. In [4] Axtell studied the zero-divisor of a commutative ring. The zero divisor graph of a commutative ring has been studied extensively by several authors [5–7].LetG be a simple connected undirected graph with the vertex set VG.LetΓR be a notation of the zero-divisor graph of a commutative ring R whose vertices are the nonzero zero-divisors of R i.e, Z∗R, with r1 and s1 adjacent if r1≠s1 and r1s1=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 defined the sum of all distances between vertices of the graph and is denoted byWG.We assume in this article the annihilator of M is annM=r∈R\0∣∀m∈M:rm=0.


## 2. Wiener Index of the GraphΓR+M

In this section, we compute the wiener index of the zero-divisor graphΓR+M.Al-Labadi M in [10], characterized the zero-divisor graph of idealization ring when R is an integral domain.Theorem 1.
LetRbe an integral domain andMbe anR−module withM=2. Then we have the following:(1)
IfannM=0, then the wiener index of the graphΓR+MisWΓR+M=0.(2)
IfannM≠0, then the wiener index of the graphΓR+MisWΓR+M=8annZ2.Proof.
We have the following:(1)
IfannM=0, then by [10] we have R≅Z2 i.e, ΓZ2+Z2=0,1 is an empty graph.(2)
IfannM≠0, then ΓR+Z2=0,1,ri,0,ri,1:ri∈annZ2. So, d0,1,ri,ti=1 and dri,t1,rj,t2=2 for all ri,rj∈annM, t1 and t2∈Z2. Hence, the wiener index of the graph ΓR+Z2 is WΓR+M=2annZ2+6annZ2=8annZ2. See, Figure 1.Figure 1
annZ2=1.Theorem 2.
LetR be an integral domain and M be an R module with M=3. Then we have the following:(1)
IfannM=0, then the wiener index of the graph ΓR+M is WΓR+M=2.(2)
IfannM≠0, then the wiener index of the graph ΓR+M is WΓR+Z3=24annZ3+2.Proof.
We have the following:(1)
IfannM=0, then by [10] we have R≅Z3 i.e, ΓR+Z3=0,1,0,2 that is WΓR+M=2.(2)
IfannM≠0, then ΓR+Z3=0,1,0,2,r,m:r∈annZ3,m∈Z3. So, d0,1,ri,t1 and dri,t1,rj,t2=2 for all ri,rj∈annM. Hence, the wiener index of the graph ΓR+M is WΓR+M=23annZ3+1+3×6annZ3=24annZ3+2. See, Figure 2.Figure 2
annZ3=1.Theorem 3.
LetR be an integral domain and M≥4 be an R− module. Then the wiener index of the graph ΓR+M is WΓR+M=M−1M−2,ifannM=0,M−1M−2+M‖annM+M‖annMM−1+2|M‖−1,ifannM≠0..Proof.
We have the following:(1)
IfannM=0, then by [10] we have ΓR+M=0,t:t∈M that is a complete graph. So, the wiener index of the graph ΓR+M is WΓR+M=M−1M−2.(2)
IfannM≠0, then by [10] we have ΓR+M=0,t,r,m:r∈annMandt,m∈M. So, d0,1,ri,t=1 and dri,t1,rj,t2=2 for all ri,rj∈annM. Hence, the wiener index of the graph ΓR+M is WΓR+M=M−1M−2+M‖annM+M‖annMM−1+2|MannM|−1.


## 3. Wiener Index of the GraphΓZN+ZM

In this section, we compute the wiener index of the graphΓZN+ZM, where ZN+ZM is called the idealization ring.[9] Al-Labadi Manal presented the zero-divisors graph of idealization ring ZN+ZM.First, we compute the wiener index of the graphΓZN+ZM, when N=p and M=p where p is a prime number.Theorem 4.
IfM=pandN=pwherep>2is a prime number, then wiener index of the graph isWΓZN+ZM=p−1p−2.Proof.
IfM=p and N=p where p>2 is a prime number, then we have ΓZp+Zp=0,t:t∈Zp∗=0,1,0,2,…,0,p−1. So, dvi,vj=1 for all vi,vj∈ΓZp+Zp. Hence, the wiener index of the graph ΓZp+Zp is WΓZp+Zp=∑i=1p−1dvi,vj=p−1p−2.Theorem 5.
IfM=pandN=p2wherep>2is a prime number, then the wiener index of the graphΓZp2+ZpisWΓZp2+Zp=p−1+pp−1pp−1+p−2.Proof.
IfM=p and N=p2 where p>2 is a prime number, then we have ΓZp2+Zp=0,m:m∈Zp∗∪kp,t,t∈Zp,gcdk,p=1. So, the graph is a complete graph. Hence, the wiener index of the graph ΓZp2+Zp is WΓZp2+Zp=p−1+pϕppϕp+p−1−1=p−1+pp−1pp−1+p−2.
Now, we characterize the sets of the zero-divisor graph for the following.
ForN=pα and M=p where p is a prime number and α>2.
L0=0,10,2,…,0,p−1 and L0=p−1.
Ls=kps,t:t∈Zp,gcdk,pα−s=1 for 1≤s≤α−1 and Ls=pϕpα−s, where ϕ is an Euler function.Theorem 6.
IfM=pandN=pαwherepis a prime number andα>2, then the wiener index of the graph isWΓZpα+Zp=∑k=0α−1βk,βk=∑vj∈Ljvi≠vj∑i=1Lkdvi,vjwhere0≤k≤α−1.Proof.
LetM=p and N=pα, α>2. Then the zero divisor graph of idealization ring Zpα+Zp is Z∗Zpα+Zp=0,m:m∈Zp∗∪Lj,j≥1.
For any vertexvi∈L0, let vj∈Lj, 0≤j≤α−1. Then we can define:
β0=∑vj∈Ljvi≠vj∑i=1L0dvi,vj.
Thenβ0=p−1∑j=1α−1Lj+L0−1. So,(1)β0=p−1∑j=1α−1pϕpα−j+p−2⋯.
For any vertexvi∈L1, let vj∈Lj, 0≤j≤α−1. Then we can define:
β1=∑vj∈Ljvj≠vi∑i=1L1dvi,vj.
Then(2)β1=L1L0+Lα−1+2∑j=1,j≠0,α−1α−2Lj⋯.
And so on, for any vertexvi∈Lα/2, let vj∈Lj, 0≤j≤α−1. Then we can define:
βα/2=∑vj∈Ljvj≠vi∑i=1Lα/2dvi,vj.
Then(3)βα/2=∑i=0α/2Lα−i+2∑j=1α−α/2+1LjLα/2⋯.
For any vertexvi∈Lα/2, let vj∈Lj. Then we can define:
βα/2=∑vj∈Ljvj≠vi∑i=1Lα/2dvi,vj.
Then(4)βα/2=∑i=0α/2Lα−i+2∑j=1α−α/2+1LjLα/2⋯.
Hence from (1)–(3) and (4) the wiener index of the graph is WΓZpα+Zp=∑k=0α−1βk.
Now, we characterize the sets of the zero-divisor graph for the following.
ForN=p1p2 and M=p1 where p1 and p2 are a prime numbers.
L0=0,1,0,2,…,0,p1−1 and L0=p1−1.
L1=kp1,t:t∈Zp1,gcdk,p2=1 and Lp1=p1ϕp2, where ϕ is an Euler function i.e, L1=p1p2−1.
L2=kp2,t:t∈Zp1,gcdk,p1=1 and Lp2=p1ϕp1, where ϕ is an Euler function i.e, L2=p1p1−1.
L1∗=kp1,0:gcdk,p2=1 and L1∗=ϕp2, where ϕ is an Euler function, i.e, L1∗=p2−1.Theorem 7.
IfM=p1andN=p1p2wherep1andp2are a prime numbers, then wiener index of the graph isWΓZp1p2+Zp1=∑k=02βk,βk=∑vj∈Ljvi≠vj∑i=1Lkdvi,vjwherek=0,1and 2.Proof.
LetM=p1 and N=p1p2 where p1 and. p2. are prime numbers. Then the zero divisor graph of idealization ring Zp1p2+Zp1 is Z∗Zp1p2+Zp1=0,m:m∈Zp1∗∪L1∪L2.
For any vertexvi∈L0, let vj∈Lj for all j∈0,1,2. Then we can define
β0=∑vj∈Ljvi≠vj∑i=1L0dvi,vj. Then(5)β0=p1−2p1−1+p1ϕp1p2−1+2p1p1−1p1−1⋯.
For any vertexvi∈L1, let vj∈Lj for all j∈0,1,2. Then we can define
β1=∑vj∈Ljvi≠vj∑i=1L1dvi,vj. Then (6)β1=p1p2−1p1−1+2p1p2−1p1p2−1=1+2p1p2−1p1p1−1….
For any vertexvi∈L2, let vj∈Lj for all j∈0,1,2. Then we can define
β2=∑vj∈Ljvi≠vj∑i=1L2dvi,vj. Then(7)β2=p1p1−1p1−1+2p1p1−1p1p1−1−1+p1p1−1p2−1+p1p1−1p2−1p1−1….
Hence, from(5), (6) and (7) the wiener index of the graph is WΓZp1p2+Zp1=∑k=02βk.


## 4. When is the GraphΓR+M Hamiltonian?

In this section, we determine when the graphΓR+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 whenR is an integral domain.Lemma 1.
LetRbe an integral domain andMbe anR−module withM=2.Then the graphΓR+Mis not a Hamiltonian graph.Proof.
We have the following:(1)
IfannM=0, then by [10], we have R≅Z2 i.e, ΓZ2+Z2=0,1 is an empty graph which is not a Hamiltonian graph.(2)
IfannM≠0, then ΓR+Z2=0,1,ri,0,ri,1,ri∈annZ2. So, can not find a cycle between a vertices. Then is not a Hamiltonian graph.Lemma 2.
LetR be an integral domain and M be an R− module with M=3. Then the graph ΓR+M is not a Hamiltonian graph.Proof.
We have the following:(1)
IfannM=0, then by [10] we have R≅Z3 i.e, ΓR+Z3=0,1,0,2 that is not a Hamiltonian graph.(2)
IfannM≠0, then ΓR+Z3=0,1,0,2,r,m:r∈annZ3,m∈Z3. So, we can not find a cycle between all vertices in the graph ΓR+M that the graph ΓR+M is not a Hamiltonian graph.
Now, we generalize the result by the following theorem.Theorem 8.
LetRbe an integral domain andM≥4be anR−module. Then the graphΓR+MisHamiltonian,ifannM=0,not Hamiltonian,ifannM≠0..Proof.
We have the following:(1)
IfannM=0, then by [10] we have ΓR+M=0,t:t∈M that is a complete graph. So, the graph ΓR+M is Hamiltonian graph.(2)
IfannM≠0, then by [10] we have ΓR+M=0,t,r,m:r∈annMandt,m∈M. So, we can not find the cycle between all vertices in the graph ΓR+M that is not a Hamiltonian graph.


## 5. When is the GraphΓZN+ZM Hamiltonian?

In this section, we determine when the graphΓZN+ZM is a Hamiltonian graph.Theorem 9.
IfM=pandN=pwherep>2is a prime number, then the graphΓZN+ZMis a Hamiltonian graph.Proof.
IfM=p and N=p where p>2 is a prime number, then we have ΓZp+Zp=0,t:t∈Zp∗=0,1,0,2,…,0,p−1 which is a complete graph. So, the graph ΓZp+Zp is a Hamiltonian graph.Theorem 10.
IfM=pandN=p2wherep>2is a prime number, then the graphΓZp2+Zpis a Hamiltonian graph.Proof.
IfM=p and N=p2 where p>2 is a prime number, then we have ΓZp2+Zp=0,m:m∈Zp∗∪L1. So, the graph is a complete graph. Hence, the graph ΓZp+Zp is a Hamiltonian graph.
Now, we investigate whenN=pα and M=p where p is a prime number and α>2.Theorem 11.
IfM=pandN=pαwhereα>2andpis a prime number, then the graphΓZpα+Zpis not a Hamiltonian graph.Proof.
IfM=p and N=pα where where α>2 and p is a prime number, then we have ΓZpα+Zp=0,m:m∈Zp∗∪Li,i≥1. So, the number of connected components of graph ΓZpα+Zp−M is not less than of the order M, by theorem in 7 where M=L0∪Lα/2∪Lα/2+1∪⋯Lα−1. Hence, the graph ΓZpα+Zp is not a Hamiltonian graph.Theorem 12.
IfM=p1andN=p1p2wherep1andp2are a prime numbers, then the graphΓZp1p2+Zp1is not a Hamiltonian graph.Proof.
LetM=p1 and N=p1p2 where p1 and p2 are prime numbers. Then the zero divisor graph of idealization ring Zp1p2+Zp1 is Z∗Zp1p2+Zp1=0,m:m∈Zp1∗∪L1UL2.
For any vertexvi∈L2 is adjacent only to any other vertex in L1∗ and any vertex in L1\L1∗ is adjacent only to vertex in L0 . Hence we can not find a cycle between all vertices in the graph ΓZp1p2+Zp1 that is not a Hamiltonian graph. See, Figure 3.Figure 3
Z6 (+) Z3


## 6. Clique Number of the GraphΓR+M

In this section, we compute the clique number of the graphΓR+M.Definition 3.
A graphGis called a maximal clique if there is no clique with more vertices.Definition 4.
The clique number of the graphGis the number of vertices in a maximum clique in the graphGwhich is denoted byωG.Theorem 13.
Let the ringRbe an integral domain andMbe anR−module withM=2. Then we have the following:(1)
IfannM=0, then the clique number of the graph ΓR+M is ωΓR+M=1.(2)
IfannM≠0, then the clique number of the graph ΓR+M is ωΓR+M=1Proof.
We have the following:(1)
IfannM=0, then by [10] we have R≅Z2 i.e., ΓZ2+Z2=0,1 and ωR+M2=1.(2)
IfannM≠0, then by [10] we have R≅Z2ΓR+Z2=0,1,ri,0,ri,1:ri∈annZ2. So, d0,1,ri,ti=1 and dri,t1,rj,t2=2 for all ri,rj∈annM, t1 and t2∈Z2. Hence, the clique number of the graph ΓR+Z2 is ωΓR+M=1. See, Figure 4.Figure 4
annZ2=1.Theorem 14.
LetRbe an integral domain andMbe anR−module withM=3. Then we have the following:(1)
IfannM=0, then the clique number of the graph ΓR+M is ωΓR+M=2.(2)
IfannM≠0, then the clique number of the graph ΓR+M is ωΓR+M=2.Proof.
We have the following:(1)
IfannM=0, then by [10] we have R≅Z3 i.e, ΓR+Z3=0,1,0,2 that is ωΓR+M=2.(2)
IfannM≠0, then by [10] we have R≅Z3 i.e, ΓR+Z3=0,1,0,2,r,m:r∈annZ3,m∈Z3. So, d0,1,ri,t=1 and dri,t1,rj,t2=2 for all ri,rj∈annM. Hence, the clique number of the graph ΓR+M is ωΓR+M=2. See, Figure 5.Figure 5
annZ3=1.Theorem 15.
LetRbe an integral domain andM≥4be anR−module. Then the clique number of the graphΓR+MisωΓR+M=M−1.Proof.
IfannM=0, then by [10] we have ΓR+M=0,t:t∈M that is a complete graph. So, the clique number of the graph ΓR+M is ωΓR+M=M−1.


## 7. Conclusion and Questions

In this section, we give the conclusion of the properties of the zero-divisor graph.Theorem 16.
LetRbe an integral domain andMbe anR-module. ThenWΓR+M=0,ifannZ2=0,M=2,8annZ2,ifannM≠0,M=2,2,ifannZ3=0,M=3,24annZ3+2ifannM≠0,M=3.Theorem 17.
LetRbe an integral domain andMbe anR-module,M≥4. ThenWΓR+M=M−1M−2,ifannM=0,M−1M−2+M‖annM+M‖annMM−1+2M‖annM−1,ifannM≠0.Theorem 18.
LetZN+ZMbe the ring of the idealization ring. ThenWΓZN+ZM=p−1p−2,ifN=M=p,p−1+pp−1pp−1+p−2,ifM=p,N=p2.Now, when the zero-divisor graph is a Hamiltonian graph.Theorem 19.
LetRbe an integral domain. Then the zero-divisor graphZN+ZMis a Hamiltonian graph ifannM=0andM≥4.Theorem 20.
LetZN+ZMbe the ring of the idealization ring. Then the zero-divisor graphΓZN+ZMis a Hamiltonian graph ifN=porN=p2.The clique number for the zero-divisor graphΓR+M when R is an integral domain.Theorem 21.
LetRbe an integral domain andMbe anR−module. Then the clique number of the graphΓR+MisωΓR+M=M−1.In the future work can be ask the following questions:(1)
What is the clique number for the zero-divisors graph of idealization ringΓR+M when R is not integral domain ?(2)
What is the geodetic number of the zero-divisors graph of idealization ring and the complement ?



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