Intelligent Automation & Soft Computing
DOI:10.32604/iasc.2021.019391
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Article

Radio Labeling Associated with a Class of Commutative Rings Using Zero-Divisor Graph

Azeem Haider1,*, Ali N.A. Koam1 and Ali Ahmad2

1Department of Mathematics, College of Sciences, Jazan University, Jazan, Saudi Arabia
2College of Computer Sciences and Information Technology, Jazan University, Saudi Arabia
*Corresponding Author: Azeem Haider. Email: aahaider@jazanu.edu.sa
Received: 11 April 2021; Accepted: 12 May 2021

Abstract: Graph labeling is useful in networks because each transmitter has a different transmission capacity to send or receive wired or wireless links. An interference of signals can occur when transmitters that are close together receive close frequencies. This problem has been modeled mathematically in the radio labeling problem on graphs, where vertices represent transmitters and edges indicate closeness of the transmitters. For this purpose, each vertex is labeled with a unique positive integer, and to minimize the interference, the difference between maximum and minimum used labels has to be minimized. A radio labeling for a graph G=(V(G),E(G)) is a function γ from the set of vertices V(G) to the set of positive integers satisfying the condition d(x,y)+|γ(x)−γ(y)|≥1+diam(G) , where d(x,y) is the shortest distance between two distinct vertices x,y∈V(G) , and diam(G) is the diameter of the graph G. The minimum span of a radio labeling for G is called the radio number of G. Because the problem of finding radio labeling appears to be difficult in general, many particular cases have been studied. Let R be a commutative ring with nonzero identity, and Z(R) its set of (nonzero) zero-divisors. The zero-divisor graph of a ring R is the graph Γ(R) with vertex set V(Γ(R))=Z(R) and edge set E(Γ(R))={(x,y):x⋅y=0} . In this paper, we investigate the radio number for an associated zero-divisor graph, Γ(Zpn) . The study provides some combinatorial properties associated with commutative rings and can be useful for the structures of network communication problems.

Keywords: Radio labeling; radio number; distances in graph; zero divisor graph; commutative ring

1  Introduction

Antennas transmit and receive different frequencies of electromagnetic waves, such as radio waves. By tuning in a radio, we receive signals to access particular frequencies. Each radio station is assigned to a distinct channel. When two radio stations are near each other, the difference between their assigned channels must be greater than a specific number to avoid interference. The task of allocating channels to transmitters is known as channel assignment (CA).

The CA model was introduced in 1980 by William Hale [1]. CA is generally modeled as a graph coloring and labeling representation, where transmitters are represented as nodes (vertices) of the graph. When two nodes are adjacent, their transmitters are close. The labels assigned to nodes are of the channels of the transmitters, where for each pair of labels, there must be an acceptable distance between their nodes. This study aims to find a suitable labeling to minimize the range (span) of the channels.

Let G be a simple and connected graph. The degree of a vertex x∈V(G) is the number of vertices adjacent to x , and is denoted as dG(x) . The shortest distance between two vertices x,y∈V(G) is denoted by d(x,y) , and the maximum value of d(x,y) in G is called the diameter of G, denoted as diam(G) . Radio labeling of G [2,3], also known as multi-level distance labeling, is a function γ:V(G)→N for which the following condition holds for any two distinct vertices x and y :

d(x,y)+|γ(x)−γ(y)|≥1+diam(G), (1)

which is referred to as a radio condition.

We denote by S(G,γ) the set of consecutive integers {m,m+1,…,M} , where m=minx∈V(G)γ(x) , and M=maxx∈V(G)γ(x) is the span of γ , denoted by span(γ)=1+M−m .

The minimum span of a radio labeling for G is called the radio number of G , denoted by rn(G) . A radio labeling γ of G with span(γ)=rn(G) will be called the optimal radio labeling for G . Radio labeling is an interesting graph labeling problem that is the subject of much research. It is complicated to determine the radio number for a general graph. The radio problem is NP-hard, even for a graph with a small diameter, and as a rule, its complication is still unknown [4]. Due to this, researchers have studied this problem, and even for some known families of the graph, the problem is shown to be complex [2,5–12].

2  Applications of Zero-divisor Graphs

The zero-divisor graph over a commutative ring was introduced in 1988 by Beck [13], who discussed the coloring of such graphs.

The interdisciplinary research in algebraic graph theory is excelling, and associated applications are benefiting from such desk research. The study conducted in [14] and [15] serves as an interesting survey to find the relation between the ring-theoretic properties and graph-theoretic properties of Γ(G) . This study deals with techniques that vary from simple computations to sophisticated ring theory, and in many cases, all the rings or graphs satisfy a certain property. We raise two questions:

     i)  Could rings with certain theoretic properties have the same physical structures and graphical properties, or vice versa?

    ii)  Is it possible to determine Γ(R) such that Γ(R)≅G ?

Redmond [16,17] provided all graphs up to 14 vertices that can be realized as the zero-divisor graph of a commutative ring with identity, listed all rings (up to isomorphism) that produce these graphs, and created an algorithm to find all commutative reduced rings with identity (up to isomorphism) that give rise to a zero-divisor graph on n vertices for any n≥1 . A question that naturally arises when studying zero-divisor graphs is whether they are unique. There are some applications and relationships between algebraic theory and chemical graph theory [18,19]. Recent work has been performed on the radio numbers of different algebraic structures [20,21].

Let p be a prime number, and Γ(Zpn) a zero-divisor graph of the commutative rings Zpn . We investigate the radio number of zero divisor graphs Γ(Zpn) for any positive integer n and prime number p.

3  Main Results

Let R be a commutative ring. A nonzero element a∈R is called a zero-divisor of R if there exists another nonzero element b∈R such that a.b=0. We denote the set of all zero divisors in a ring R by Z(R) . Assume that R is a ring with identity 1 . Then an element u∈R is called a unit in R if there exists an element v∈R such that u.v=1. We denote the set of all units in R by U(R). A unit element in a ring R cannot be a zero divisor. Similarly, a zero-divisor in R cannot be a unit but a.u∈Z(R) for any a∈Z(R) and u∈U(R) .

We consider rings of the form R=Zk for a fixed positive integer k, where a nonzero element is either a unit or a zero-divisor. More precisely, an element a∈Zk∖{0} is a zero divisor if and only if gcd(a,k)≠1 , and an element u∈Zk∖{0} is a unit if and only if gcd(u,k)=1.

For a fixed prime p and a fixed positive integer n , we consider the ring R=Zpn. An element a∈Zpn∖{0} is a zero-divisor if and only if p divides a . It is easy to see that the set of zero-divisors Z(Zpn)=∐i=1n−1Zi, where each Zi={u.pi:uisaunitinZpn} contains those elements of Zpn that are multiples of pi but not of pi+1 . Therefore, |Zi|=pn−i−pn−i−1 for each i=1,2,…,n−1 , and hence |Z(Zpn)|=∑i=1n−1|Zi|=pn−1−1.

We associate a zero-divisor graph Γ(Zpn) to a ring Zpn with vertex set V(Γ(Zpn))=Z(Zpn) ; note that we consider a zero-divisor to be a nonzero element, so 0∉V(Γ(Zpn)). The degree of each vertex in Zi is shown in the following theorem.

Theorem 3.1 Let Γ(Zpn) be a zero-divisor graph of Zpn . Then,

dZi(x)={pi−1,for1≤i≤⌈n2⌉−1pi−2,for⌈n2⌉≤i≤n−1

Proof. For any vertex x∈Zi , we have x.y=0 if and only if y∈Zj for j≥n−i . For 1≤i≤⌈n2⌉−1 , we get dZi(x)=|∐j=n−in−1Zj|=∑j=n−in−1|Zj|=pi−1. . For ⌈n2⌉≤i≤n−1 , we get dZi(x)=|∐j=n−in−1Zj−{x}|=∑j=n−in−1|Zj|−1=pi−1−1=pi−2.

Using the handshaking lemma, after simplification, we obtain the size of Γ(Zpn) in the following theorem.

Theorem 3.2 For n≥2 and prime number p , the size of Γ(Zpn) is

12{∑x∈V(Γ(Zpn))d(x)}=12{pn−1(np−n−p)−pn−⌈n2⌉+2} , except n=p=2.

We know that the radio number of complete graph Kn is n . From the definition, it can be seen that the zero-divisor graph of Γ(Zpn) is a complete graph Kp−1 for n = 2. Therefore, the following theorem holds.

Theorem 3.3 Let Γ(Zpn) be a zero-divisor graph of Zpn . Then rn(Γ(Zpn))=p−1 for n=2.

In the next theorem, we determine the radio number of a zero-divisor graph of Zpn for n=3.

Theorem 3.4 Let p≥2 be a prime number, and Γ(Zp3) a zero-divisor graph of Zp3 . Then rn(Γ(Zp3))=p2+p−2.

Proof. For simplicity, we partition the vertex set of Γ(Zp3) into two mutually disjoint sets, Z1={u.p:u≠k1.p,1≤k1≤p−1}={ui:1≤i≤p2−p} and Z2={u.p2:u=k2.p,1≤k1≤p−1}={wj:1≤j≤p−1} . According to the definition of the zero-divisor graph, the vertices of the set Z1 are not adjacent, and the vertices of the set Z2 are adjacent. This means that d(u1,u2)≠1 and d(w1,w2)=1 for u1,u2∈Z1 and w1,w2∈Z2 . Also, all the vertices of set Z1 are adjacent to each vertex of set Z2 . Hence, d(u,w)=1 ∀u∈Z1,w∈Z2 and |Z1|=p2−p , |Z2|=p−1 . From this, we observe that d(u1,u2)=2. This indicates that the diameter of Γ(Zp3) is 2. Therefore, the zero-divisor graph Γ(Zp3) must satisfy the radio condition defined in Eq. (1),

d(x,y)+|γ(x)−γ(y)|≥3,forx,y∈V(Γ(Zp3)). (2)

Since d(u1,u2)=2 , there are no forbidden values between the vertices of set Z1 . Since d(w1,w2)=1 , there are p−2 forbidden values between the vertices of set Z2 . Also, d(u,w)=1 , which gives only one forbidden value between the vertices of sets Z1 and Z2. Therefore, there is a total of p−1 forbidden values in zero-divisor graph Γ(Zp3). Hence, the lower bound of the radio number of zero-divisor graph Γ(Zp3) is

rn(Γ(Zp3))≤|V(Γ(Zp3))|+p−1=p2+p−2. (3)

To obtain the upper bound of the radio number of zero-divisor graph Γ(Zp3) , we define the radio labeling γ1:V(Γ(Zp3)→{1,2,3,…,p2+p−2} as γ1(ui)=i for 1≤i≤p2−p , and γ1(wj)=p2−p+2j for 1≤j≤p−1. Without loss of generality, assume that for any two vertices us,ut∈Z1 , ws,wt∈Z2 , |γ1(us)−γ1(ut)|≥1 , |γ1(ws)−γ1(wt)|≥2 , and |γ1(uj)−γ1(wi)|=p2−p+2j−i≥2. This demonstrates that the radio labeling γ1 satisfies the radio condition (2) for zero-divisor graph Γ(Zp3) . Therefore,

rn(Γ(Zp3))≥p2−p+2(p−1)=p2+p−2. (4)

Combining Eqs. (3) and (4), we obtain the required result. This completes the proof.

Theorem 3.5 Let p be a prime number, and Γ(Zp4) a zero-divisor graph of Zp4 . Then rn(Γ(Zp4))=p3+p2−3.

Proof. For simplicity, we partition the vertex set of Γ(Zp4) into three mutually disjoint sets: Z1={u.p:u≠k1.p2,1≤k1≤p2−1}={xi:1≤i≤p3−p2} , Z2={u.p2:u≠k2.p,1≤k2≤p3−p2+p−1}={yj:1≤j≤p2−p} , and Z3={u.p3:u≠k3.p,1≤k3≤p3−p}={zt:1≤t≤p−1} . According to the definition of the zero-divisor graph, the vertices of set Z1 are not adjacent, and the vertices of sets Z2 and Z3 are adjacent. This means that d(x1,x2)≠1 and d(y1,y2)=d(z1,z2)=1 for x1,x2∈Z1 , y1,y2∈Z2 , and z1,z2∈Z3 . Additionally, all the vertices of sets Z1 and Z2 are adjacent to each vertex of set Z3 . This means that d(x,z)=d(y,z)=1 ∀x∈Z1,y∈Z2,z∈Z3 , and |Z1|=p3−p2 , |Z2|=p2−p , |Z3|=p−1 . This implies that |V(Γ(Zp4))|=p3−1 . From the above discussion, it is observed that d(x1,x2)=d(x,y)=2. This shows that the diameter of Γ(Zp4) is 2. Therefore, the zero-divisor graph Γ(Zp4) must satisfy the radio condition defined in Eq. (1), i.e.,

d(u,v)+|γ(u)−γ(v)|≥3,foru,v∈V(Γ(Zp4)). (5)

Since d(x1,x2)=2 , there is no forbidden value between the vertices of set Z1 and d(y1,y2)=1 . Therefore, there are p2−p−1 forbidden values between the vertices of set Z2 and d(z1,z2)=1 . Similarly, there are p−2 forbidden values between the vertices of set Z3 . Since d(x,y)=2 , this shows that there is no forbidden value between the vertices of sets Z1 and Z2. Also d(x,z)=d(y,z)=1 , which gives only one forbidden value between the vertices of set Z3 and sets Z1,Z2 . Thus there are p2−p−1+p−2+1=p2−2 forbidden values in the zero-divisor graph Γ(Zp4). Hence, the lower bound of the radio number of zero-divisor graph Γ(Zp4) is

rn(Γ(Zp4))≤|V(Γ(Zp4))|+p2−2=p3+p2−3. (6)

To obtain the upper bound of the radio number of zero-divisor graph Γ(Zp4) , we define the radio labeling γ2:V(Γ(Zp4))→{1,2,3,…,p3+p2−3} as follows.

γ2(xi)=i for 1≤i≤p3−p2 , γ2(yj)=p3−p2+2j−1 for 1≤j≤p2−p , and γ2(zt)=p3+p2−2p+2t−1 for 1≤t≤p−1. For any two vertices xs,xt∈Z1 , ys,yt∈Z2 , zs,zt∈Z3 , |γ2(xs)−γ2(xt)|≥1 , |γ2(ys)−γ2(yt)|≥2 , |γ2(zs)−γ2(zt)|≥2 , |γ2(xi)−γ2(yj)|≥1 , |γ2(xi)−γ2(zt)|≥2 , and |γ2(yj)−γ2(zt)|≥2 . In addition, d(xs,xt)=d(x,y)=2 and d(xi,zt)=d(yj,zt)=d(ys,yt)=d(zs,zt)=1 . This shows that the radio labeling γ2 satisfies the radio condition (2) for zero-divisor graph Γ(Zp4) . Therefore, we arrive at

rn(Γ(Zp4))≥p3+p2−3. (7)

Combining Eqs. (6) and (7), we obtain the required result. This completes the proof.

The above theorems lead us to establish general results related to the radio number of zero-divisor graphs associated to rings Zpn for n≥5 .

In the following proposition, we determine the lower bound of the radio number for a zero-divisor graph of Zpn for n≥5 .

Proposition 3.6 Let p be a prime number and n≥5 . Then rn(Γ(Zpn))≥pn−1+p⌊n2⌋−3 .

Proof. From the above discussion, we have V(Γ(Zpn))=Zi={u.pi:uisaunitinZpn} . This means that it contains those elements of Zpn which are multiples of pi but not of pi+1 , which implies that |Zi|=pn−i−pn−i−1 for 1≤i≤n−1 . Therefore, V(Γ(Zpn))=pn−1−1 . Let dZi(x) denote the degree of a vertex x in set Zi , and d(Zi,Zj) the distance between the vertices of sets Zi and Zj . For any vertex x1i∈Zi , we have dZi(x1i)=pi−1 for 1≤i≤⌈n2⌉−1 and dZi(x1i)=pi−2 for ⌈n2⌉≤i≤n−1 . Additionally,

d(x1i,x2j)={1,ifi=jand⌈n2⌉≤i≤n−12,ifi=jand1≤i≤⌈n2⌉−1 d(x1i,x2j)={2,ifi≠jand1≤i,j≤⌈n2⌉−11,ifi≠jand⌈n2⌉≤i,j≤n−1 d(x11,x2j)={2,if2≤j≤n−21,ifj=n−1,

and d(x1i,x2j)=1 for 2≤i≤⌈n2⌉−1 and ⌈n2⌉≤j≤n−1 . This implies that diam(Γ(Zpn))=2 . Any radio labeling γ of Γ(Zpn) must satisfy the following radio condition:

d(x1i,x2j)+|γ(x1i)−γ(x2j)|≥diam(Γ(Zpn))+1=3,

for any distinct vertices x1i,x2j∈V(Γ(Zpn)). We now count the forbidden values for Γ(Zpn) . If d(x1i,x2j)=2 , then it is possible to assign consecutive labels between those vertices. This means there is no forbidden value between them. Therefore, for n -even, there are no forbidden values between the vertices of Zi ( 1≤i≤n2−1 ) and Zj ( n2≤j≤n−1 ), and for n -odd, there are no forbidden values between the vertices of Zi ( 1≤i≤n−12 ) and Zj ( n+12≤j≤n−1 ). Now, if d(x1i,x2j)=1, then |γ(x1i)−γ(x2j)| must be greater than 2. This means there must be a forbidden value between those vertices. Therefore, for n -even, there are ∑i=n2n−1|Zi|−1 forbidden values between the vertices of Zi ( n2≤i≤n−1 ), and for n -odd, there are ∑i=n+12n−1|Zi|−1 forbidden values between the vertices of Zi ( n+12≤i≤n−1 ). By adding the forbidden values to the order of the graph, we obtain the total number of labels.

Hence, for n -even,

rn(Γ(Zpn))≥|V(Γ(Zpn))|+∑i=n2n−1|Zi|−1=∑i=1n2−1|Zi|+2∑i=n2n−1|Zi|−1=pn−1+p⌊n2⌋−3, (8)

and for n -odd,

rn(Γ(Zpn))≥|V(Γ(Zpn))|+∑i=n+12n−1|Zi|−1=∑i=1n−12−1|Zi|+2∑i=n+12n−1|Zi|−1=pn−1+p⌊n2⌋−3. (9)

Combining Eqs. (8) and (9), we arrive at

rn(Γ(Zpn))≥pn−1+p⌊n2⌋−3. (10)

This completes the proof.

The next proposition determines the upper bound of the radio number for a zero-divisor graph of Zpn for n≥5 .

Proposition 3.7 Let p be a prime number and n≥5 . Then rn(Γ(Zpn))≤pn−1+p⌊n2⌋−3 .

Proof. We provide a radio labeling of Γ(Zpn) with span pn−1+p⌊n2⌋−3 , which implies that rn(Γ(Zpn))≤pn−1+p⌊n2⌋−3 .

The radio labeling γ:V(Γ(Zpn))→Z+ is defined as follows.

Case 1: n -even

For xji∈Zi , where |Zi|=pn−i−pn−i−1 and |Z0|=0 ,

γ(xji)={j+∑s=1i|Zs−1|,if1≤j≤|Zi|and1≤i≤n2−12j−1+∑s=1n2−1|Zs|+2∑s=n2i|Zs−1|−2|Zn2−1|,if1≤j≤|Zi|andn2≤i≤n−1

Case 2: n -odd

For xji∈Zi , where |Zi|=pn−i−pn−i−1 and |Z0|=0 ,

γ(xji)={j−1+∑s=1i|Zs−1|+2∑s=n+12n−1|Zs|,if1≤j≤|Zi|and1≤i≤n−122j−1+2∑s=n+12n−1|Zs|−2∑s=n+12i|Zs|,if1≤j≤|Zi|andn+12≤i≤n−1

From case 1, it can be seen that Γ(Zpn) attains an upper bound if and only if i=n−1 and j=|Zn−1| , i.e.,

2|Zn−1|−1+∑s=1n2−1|Zs|+2∑s=n2n−1|Zs−1|−2|Zn2−1| =2(p−1)−1+∑s=1n2−1|Zs|+2∑s=n2+1n−1|Zs−1|+2|Zn2−1|−2|Zn2−1| =2p−3+∑s=1n2−1|Zs|+2∑s=n2n−2|Zs| =2p−3+∑s=1n−2|Zs|+∑s=n2n−2|Zs| =2p−3+pn−1−pn−n+2−1+pn2−p =pn−1+pn2−3=pn−1+p⌊n2⌋−3.

Similarly, from case 2, it can be seen that Γ(Zpn) attains an upper bound if and only if i=n−12 and j=|Zn−12| , i.e.,

|Zn−12|−1+∑s=1n−12|Zs−1|+2∑s=n+12n−1|Zs| =|Zn−12|−1+∑s=1n−32|Zs|+2∑s=n+12n−1|Zs| =−1+∑s=1n−12|Zs|+2∑s=n+12n−1|Zs| =−1+∑s=1n−1|Zs|+∑s=n+12n−1|Zs| =−1+pn−1−1+pn−12−1=pn−1+pn−12−3=pn−1+p⌊n2⌋−3.

One can see that for both cases, the span of γ is equal to pn−1+p⌊n2⌋−3.

Claim: The labeling γ is a valid radio labeling. We must show that the following radio condition holds for all pair of vertices x1i,x2j∈V(Γ(Zpn)) , where x1i≠x2j :

d(x1i,x2j)+|γ(x1i)−γ(x2j)|≥diam(Γ(Zpn))+1=3. (11)

Case 1: n -even

1: Consider the pair (x1i,x2j) with 1≤i,j≤n2−1 . Note that d(x1i,x2j)=2 and |γ(x1i)−γ(x2j)|≥1 . Hence, radio Eq. (11) is satisfied.

2: Consider the pair (x1i,x2j) with n2≤i,j≤n−1 . We have d(x1i,x2j)=1 and |γ(x1i)−γ(x2j)|≥2 . Hence, radio Eq. (11) is satisfied.

3: Consider the pair (x1i,x2j) with i=1,j=n−1 . We have d(x1i,x2j)=1 and |γ(x1i)−γ(x2j)|≥2 . Hence, radio Eq. (11) is satisfied.

4: Consider the pair (x1i,x2j) with i=1,n2≤j≤n−2 . We have d(x1i,x2j)=2 and |γ(x1i)−γ(x2j)|≥2 . Hence, radio Eq. (11) is satisfied.

5: Consider the pair (x1i,x2j) with 2≤i≤n2−2,n2≤j≤n−1 . We have d(x1i,x2j)=1 and |γ(x1i)−γ(x2j)|≥2 . Hence, radio Eq. (11) is satisfied.

6: Consider the pair (x1i,x2j) with i=n2−1,j=n2 . We have d(x1i,x2j)=2 and |γ(x1i)−γ(x2j)|≥1 . Hence, radio Eq. (11) is satisfied.

7: Consider the pair (x1i,x2j) with i=n2−1,n2+1≤j≤n−1 . We have d(x1i,x2j)=1 and |γ(x1i)−γ(x2j)|≥2 . Hence, radio Eq. (11) is satisfied.

We have shown that condition (11) is satisfied for all pairs. This means that rn(Γ(Zpn))≤span(γ)=pn−1+p⌊n2⌋−3 for n -even. Similarly, it is easy to show that rn(Γ(Zpn))≤span(γ)=pn−1+p⌊n2⌋−3 for n -odd. This implies that rn(Γ(Zpn))≤span(γ)=pn−1+p⌊n2⌋−3 for n≥5. This completes the proof.

Theorem 3.8 Let p be a prime number, and n≥5 a positive integer. The radio number for a zero-divisor graph of Zpn is pn−1+p⌊n2⌋−3 , i.e., rn(Γ(Zpn))=pn−1+p⌊n2⌋−3 .

Proof. Combining Propositions 3.6 and 3.7, we obtain rn(Γ(Zpn))=pn−1+p⌊n2⌋−3 for any n≥5 .

4  Conclusion

We determined the radio number for the zero-divisor graph Γ(Zpn) of the commutative ring Zpn . In addition to the importance of the study on combinatorial properties associated with algebraic structure, these results can also be useful for circuit design and communication problems such as channel assessment.

Funding Statement: The authors received no funding for this study.

Conflict of Interest: The authors declare that they have no conflict of interest regarding the present work.

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