# Frequency Assignment Model of Zero Divisor Graph

Authors: R. Radha; N. Mohamed Rilwan
Journal: Journal of Applied Mathematics
 (2021)

---

## Abstract

Given a frequency assignment network model is a zero divisor graphΓ=V,E of commutative ring Rη, in this model, each node is considered to be a channel and their labelings are said to be the frequencies, which are assigned by the L2,1 and L3,2,1 labeling constraints. For a graph Γ, L2,1 labeling is a nonnegative real valued function f:VG⟶0,∞ such that ∣fx−fy∣≥2d if d=1 and ∣fx−fy∣≥d if d=2 where x and y are any two vertices in V and d>0 is a distance between x and y. Similarly, one can extend this distance labeling terminology up to the diameter of a graph in order to enhance the channel clarity and to prevent the overlapping of signal produced with the minimum resource (frequency) provided. In general, this terminology is known as the Lh,k labeling where h is the difference of any two vertex frequencies connected by a two length path. In this paper, our aim is to find the minimum spanning sharp upper frequency bound λ2,1 and λ3,2,1, within Δ2, in terms of maximum and minimum degree of Γ by the distance labeling L2,1 and L3,2,1, respectively, for some order η=pnq,pqr,pn where p,q,r are distinct prime and n is any positive integer.


---

## Body

## 1. Introduction

Frequency assignment problem is one of the mathematical optimization technique used in wireless communication. The frequency assignment problem (FAP) initially emerged in the 1960s. Today, the wireless telecommunication networks such as mobile network, radio, television broadcasting, satellite, and radar string our world in our palm. The development and innovation on this network become natural, essential, and unavoidable. It is clear that a number of waves produced in one second is known as frequency and is denoted asλ. For various frequencies, we can access different channels. The proximity between the channel frequencies may affect the channel clarity of one another, and at the same time, our objective is to limit the use of resources. Therefore, distance between the channels is a main determinable factor to overcome the interference as well as for a feasible solution. FCA (fixed channel assignment), DCA (dynamic channel assignment), and HCA (hybrid channel assignment) are the different types of FAP models; for further more classification, one can read the literature source [1].In 1980, Hale introduced this as a vertex colouring problem [2]. Shao et al. proposed “efficient assignment of radio channels to transmitters” in 1982 [3]. The L2,1 analogue was first proposed by Griggs and Yeh in 1992 [4], and they discussed L2,1 labeling bound for some basic graphs in terms of maximum degree Δ. According to Griggs, for a graph GV,E, an L2,1 labeling is a nonnegative real valued function f:VG⟶0,∞ such that ∣fx−fy∣≥2d if d=1 and ∣fx−fy∣≥d if d=2 where x and y are any two vertices in V and d>0 is the distance between x and y. The L2,1 labeling number of G is the smallest number such that no other label does not exceed that and is denoted by λ=λG. The canonical generalization of L2,1 labeling is Lh,k labeling where h is the difference of any two vertex frequencies connected by a two length path and λh,k is the minimum spanning labeling number.At this instance, our FAP input model is zero divisor graphΓ=V,E of a commutative ring Rη [5]. This problem was also first introduced as a colouring problem like FAP. The notion of this graph was initiated by Beck in 1988; after one decade, it was even more fined by Anderson and Livingston as an algebraic graph problem which gives more exposure to this graph. Let ZD be a set of all nonzero zero divisors of Rη, which is the vertex set VΓ. An edge between two distinct vertices u and v if and only if uv=0. Our aim of output is to attain minimum spanning frequency λ=λΓ such that no other frequency is greater than λ under the constraint of distance. Suppose, for a (frequency model) graph, Γ has a diameter two then L2,1 labeling (assignment) as same as Lh,k labeling (assignment) of Γ. Let f be a frequency function and d be a distance of v from u, on the vertex set VΓ to the frequency domain 0,∞ where fv be a frequency for any vertex (channel) v∈VΓ. Suppose f∈Ld2,1Γ, then f is a L2,1 labeling function of Γ if for all u,v∈V(1)fu−fv≥2ifdu,v=1,1ifdu,v=2.Supposef∈Ld3,2,1Γ, then f is a L3,2,1 labeling function of Γ if for all u,v∈V(2)fu−fv≥3ifdu,v=1,2ifdu,v=2,1ifdu,v=3.Note that our FAP follows the static model [1] FCA, and then, the connections remain the same ever. For a graph G, the general upper bound on λ2,1 is never much larger than Δ2 said by Griggs [4] and also he proved λG≤Δ2+2Δ. Over time, this general bound was again restricted as λG≤Δ2+Δ by Chang and Kuo [6]. In this manuscript, we determined the sharp upper bound on λ2,1 and λ3,2,1 of a zero divisor graph of a commutative ring, in terms of Δ (maximum degree of Γ) and δ (minimum degree of Γ) which also satisfies the general upper bound of Griggs and others [3, 4]. The algorithmic approach is an optimized way for solving these types of polynomial time, NP complete and network problems. Many methods have been found for constructing the algorithm for creating a zero divisor graph. Labeling scheme points the correct labeling block or a label in localized networks associated with some defined conditions. This was introduced by Breuer in 1996 [7]. We the authors mention the labeling scheme before executing every labeling function, which points the correct labeling block or a subblock of graph Γ.


## 2. Upper Bounds in terms ofΔ and δ

In this section, we have obtained the exact bound onL2,1 and L3,2,1 labeling number for some classes of zero divisor graph of finite commutative ring whose order is denoted by η. Throughout this paper, we used f as a nonnegative real valued function for labeling of Γ for some order η is pnq,pn and pqr. It is very essential to analyze the structure of the graph, and it is clear that ∣ZDRη∣=η−ϕη−1. These zero divisors are contributed by each nontrivial divisor of η. Let pn−1q,pn−2q,pn−3q,⋯,pn−n−1q,q and pn,pn−1,⋯,p2,p be the nontrivial divisors of the ring R of order η=pnq. Consider the set s2=∑i=1n−1∣mupn−iq∣, where mupn−iq is the unique multiple of pn−iq and ∣mupn−iq∣ is the contribution of vertices (zero divisors) by nontrivial divisors of η, including itself such that, for any i(3)mupn−iq=η−pn−iqpn−i.Therefore,s2=mupn−1q+mupn−2q+⋯+mupn−n−1q. Now, the problem is to climb the cardinality of s2=pn−1−1 by inductive steps
(4)mupn−1q=mpn−1q=p−1,ifi=1canonical,mupn−2q=mpn−2q−mpn−1q,ifi=2,⋮mupn−n−1q=mpn−n−1q−mpn−n−2q,ifi=n−1.By observation, we get(5)∑i=1n−1mupn−iq=p−1+pp−1+p2p−1+⋯+pn−2p−1=pn−1−1.In particular,(6)muq=mupn−nq=pn−pn−1.From equation (6), we can construct the set s4 and s4=pn−pn−1. Let s3 be a set of all vertices of ΓRη such that s3=∑i=2nmupi, where mupi is the number of unique multiple of pi, and by the previous discussion, for any i, mpi=η−pi/pi.
(7)mupn=mpn=q−1,ifi=n,mupn−1=mpn−1−mpn+mupn−1q,ifi=n−1,mupn−2=mpn−2−mpn−1+mupn−2q,ifi=n−2,⋮mupn−n−2=mp2−mp3+mup2q,ifi=2.Hence,(8)∑i=2nmupi=q−1+p0q−1p−1+⋯+pn−3q−1p−1=q−1pn−2.When the summation is taken fromi=1, we get
(9)mupn−n−1=mup=q−1pn−1−pn−2=s1let.From equations (9), (8), (6), and (5), one can find the cardinality of vertices contributed by the nontrivial divisors of η, which are mentioned initially. By adding those equations, we get pn+q−1pn−1−1 which is the number of zero divisors of a ring as well as the vertex set VΓ; in addition, the author will express the maximum degree Δ and minimum degree δ of Γ in terms of p and q such that
(10)Δ=pn−1q−2ifp<q,pn−1ifp>q,δ=p−1ifp<q,q−1ifp>q,of Γ for a ring with order pnq where p and q are distinct prime and n is any positive integer.Theorem 1.
LetΓ be a zero divisor graph of order η=pnq where p and q are distinct prime and n is any positive integer. Then, λ2,1Γ=Δ+δ where Δ and δ are maximum and minimum degrees of Γ, respectively.Proof.
Type 1. Suppose p<q.
Case 1. If n/2>1, then the labeling scheme s2Ki⟶s1⟶s3⟶s2di.
Here,Ki’s denote the vertices of complete subblock whereas di’s denote the other vertices of s2.
(11)fvs2Ki=2i−2,1≤i≤pn/2−1,fvs1i=2i+2p−3,1≤i≤pn/2−p,i+pn/2+p−3,pn/2−p+1≤i≤q−1pn−1−pn−2,fvs3i=i+pn/2+p−3+q−1pn−1−pn−2,1≤i≤q−1pn−2,fvs2di=i+pn/2+p−3+q−1pn−1,1≤i≤pn−1−pn/2.
Case 2. If n/2=1, then the labeling scheme s2⟶s1⟶s3.
(12)fvs2Ki=2i−2,1≤i≤pn/2−1,fvs2di=i+2pn/2−3,1≤i≤pn−1−pn/2−1.fvs1i=i+pn/2+pn−1−3,1≤i≤q−1pn−1−pn−2,fvs3i=i+qpn−1−pn−2q−1+pn/2−3,1≤i≤q−1pn−2.
As per requirement, we can assign the distinct labels for allv∈s4 from the same labeling domain of s1. Since ∣s1∣>∣s4∣ in addition the distance from s1 to s4 and s4 to ∣s3∣−q−1, vertices of s3 are three.
Subcase 1. If n=2, then such graphs have no vdi’s in s2.
(13)fvs2Ki=2i−2,1≤i≤pn/2−1,fvs1i=i+2pn/2−3,1≤i≤q−1pn−1−pn−2,fvs3i=i+q−1pn−1−pn−2+q−1+2pn/2−3,1≤i≤q−1pn−2.
In this subcase, the sets4 vertices are at the distance of three from s1. The analogue of this labeling allows to assign the same labeling of s1 to the vertices of s4.
Type 2. Suppose p>q.
Case 1. If n/2>1, then the labeling scheme s2Ki⟶s4⟶s2di⟶s3q−1 elements.
(14)fvs2Ki=2i−2,1≤i≤pn/2−1.
At this instance,pn/2−2 vertices of s4 can be labeled by the numbers between the two existing labels that are assigned to the vertices of s2Ki(15)fvs4i=i+2p−4,1≤i≤pn−pn−1−pn/2+2,i−2pn−pn−1−pn/2+2+1−i,pn−pn−1−pn/2+3≤pn−pn−1,fvs2di=i+pn/2+pn−pn−1−2,1≤i≤pn−1−pn/2,fvs3i=i+pn−1,1≤i≤q−1.
Case 2. If n=3, then the labeling scheme s2⟶s4⟶s3.
(16)fvs2Ki=2i−2,1≤i≤p−1,fvs2di=i+2p−9,p≤i≤pn−1−1.
At the end of this execution, there arepn/2−1 missed numbers between the existing labels of s2 which can be accessed and assigned as the labels, for some v∈s4, with the next step of scheme execution as follows:
(17)fvs4i=i+2p+pn−1−10,1≤i≤pn−pn−1−pn/2+1,i−2pn−pn−1−pn/2+1+1−i,pn−pn−1−pn/2+2≤i≤pn−pn−1,fvs3i=i+2p+pn−pn/2−8,1≤i≤q−1.
This frequency assignment system can be classified into two types along with their labeling scheme. By applying the above defined labeling function step by step as per the labeling scheme, the channel frequency least upper bound is attained.Theorem 2.
LetΓ be a zero divisor graph of a ring of order η=pnq where p and q are distinct prime and n is any positive integer. Then, for a maximum degree Δ of Γ,
(18)λ3,2,1Γ=2Δ+p−1,ifn2=1whenp<q,2Δ+pn/2−2,ifn2≥2whenp<q,2Δ+pn/2+2q−5,whenp>q.Proof.
SinceΓ is a zero divisor graph of a finite commutative ring of order η=pnq, the set of all nonzero zero divisors ZDRη is the vertex set VΓ of cardinality pn+q−1pn−1−1. Any two vertices x and y of VΓ are said to be adjacent if xy=0 to η where zero is the additive identity. The vertex set VΓ has the partition such that VΓ=s1∪s2∪s3∪s4, which are the unique multiples of p,pn−1q∪pn−2q∪⋯∪pn−n−1q, pnq∪pn−1q∪⋯∪p2,q, respectively. An element x belongs to any one of the nontrivial divisor collection ℓ such that x,η=ℓ; this will ensure the unique existence of the element in each set. An edge xy belongs to the different partition or in the same partition; the corresponding vertices are zero to at least prime factor of η. In s2, n/2 multiples of nontrivial divisor collections are adjacent to all other element of the same collection and they individually form the n/2 complete graphs; at the same time, they together form Kpn/2−1 in s2, which is denoted by vs2Ki. The remaining pn−1−pn/2 elements are not adjacent to one another, which is denoted by vs2di.
Type 1. Suppose p<q, then the labeling scheme s2Ki⟶s1⟶s3⟶s2di⟶s4.
(19)fvs2Ki=3i−31≤i≤pn/2−1,fvs1i=3p−3,i=1,ifn2=1,2i+3p−5,2≤i≤q−1pn−1−pn−2,ifn2=1,fvs1i=2i+3pn/2−2,1≤i≤q−1pn−1−pn−2,ifn2≥2,fvs3i=2i+2q−1pn−1−pn−2+3pn/2−2,1≤i≤q−1pn−2,ifn2≥2,2i+2q−1pn−1−pn−2+3p−5,1≤i≤q−1pn−2,ifn2=1,fvs2di=2i+2q−1pn−1+3pn/2−5,1≤i≤pn−1−pn/2.
It is clear that the minimum distance froms4 is three to s1 and ∣s3∣−q−1 elements of s3. For assigning labels to the elements of s4, at this instance, we can expect two possibilities ∣s1∣=∣s4∣ and ∣s1∣>∣s4∣.
If∣s1∣=∣s4∣, then
(20)fvs4i=2i+3p−6,1≤i≤pn−pn−1ifn2=1,2i+3pn/2−5,1≤i≤pn−pn−1−1ifn2≥2,2i+3pn/2+q2−6,i=pn−pn−1ifn2≥2.If ∣s1∣>∣s4∣, then
(21)fvs4i=2i+3pn/2−5if1≤i≤pn−pn−1.
Type 2. Suppose p>q, then
(22)fvs2ki=3i−31≤i≤pn/2−1,fvs4i=2i+3pn/2−2,1≤i≤pn−pn−1,fvs1i=2i+3pn/2−5,1≤i≤q−1pn−1−pn−2.
Case 1. If n=2, then
(23)fvs3i=3i+2pn−pn−1+3pn/2−2ifi=1,2i+2pn−pn−1+3pn/2−5if2≤i≤q−1.
Case 2. If n>2, then
(24)fvs2di=2i+2pn−pn−1+3pn/2−2if1≤i≤pn−1−pn/2,fvs3i=2pn+pn/2−3ifi=1,2i+2pn+pn/2−5if2≤i≤q−1,2i+2q−1pn−1−pn−2+3pn/2−5if1≤i≤q−1pn−2−1.
Thus, we have attained theL3,2,1 number λ3,2,1 of Γ which is minimum.Example 3.
Figure1 describes the FAP model which resembles a zero divisor graph of a finite commutative ring of order pnq=74×5=12,005 along with their frequency assignments. For this ring, the cardinality of zero divisors acts as a number of channels (vertices) such that ∣VΓ∣=∣ZD∣=3772. The vertices are partitioned into four separate blocks, namely, s1, s2, s3, and s4, respectively. Note that ∣s1∣=1176, ∣s2∣=6+42+294, ∣s3∣=4+24+168, and ∣s4∣=2058. The dotted subblock and the darkened subblock of s2 together form the complete graph which is denoted by s2Ki where the dotted subblock alone is adjacent with block s1. The other subblock of s2 is adjacent with both darkened and dotted subblock but not within itself. In s3, the dotted subblock alone is adjacent with s4 block. Adjacency relation between s2 and s3 blocks is shown in the figure. According to Theorem 2, it comes under the Case 2 of Type 2. By applying the respective frequencies to the vertices, one can attain the least upper frequency λ3,2,1Γ which is 2Δ+pn/2+2q−5 where Δ=pn−1.Figure 1
L3,2,1 labeling of Γ of order 74×5.A graph can be decomposed into a complete graph, and an independent set is known as a split graph. Our next frequency model is also a zero divisor graphΓ of a finite commutative ring of order η=pn, which is isomorphic to the split graph. Let s1 be a set consisting the vertices of the complete graph and the collection of independent vertices is said to be set s2. Both sets are having the cardinalities p−1 and pn−1−p, respectively. It is clear that ∣VΓ∣=∣s1∣+∣s2∣=pn−1 are the set of zero divisors of ring Rη. Therefore, the maximum degree is pn−1−2 and the minimum degree is p−1.Theorem 4.
LetΓ be a zero divisor graph of order η=pn where p is prime and n is any positive integer. Then, λ2,1Γ=Δ+δ and λ3,2,1Γ=2Δ+δ where Δ and δ denote the maximum and minimum degrees of Γ, respectively.Proof.
For this model, we discussed two different frequency distributions. Letf be a frequency distribution function which is executed in two distributions with the same labeling scheme s1⟶s2 but different distance constraints.
Frequency Assignment 1(25)fvs1i=2i−2,1≤i≤p−1,fvs2i=i+2p−3,1≤i≤p2−p.
Frequency Assignment 2(26)fvs1i=3i−3,1≤i≤p−1,fvs2i=2i+3p−5,1≤i≤p2−p.
By applying the labeling functionf in each assignment, we can obtain the L2,1 number λ2,1=p2+p−3=Δ+δ and the L3,2,1 number λ3,2,1=2p2+p−5=2Δ+δ of Γ, respectively.In a complete tripartite graph, each partite element is adjacent with a certain set of elements but no one path of length less than two does occur between those sets of elements. This structured graph is isomorphic to the zero divisor graph of a finite commutative ring of orderη=pqr where p,q,r are distinct prime. For that, p,q,r,qr,pr,pq are the nontrivial divisors of the graph. Each divisor and its multiples together form a collection of vertices of Γ that are indexed as the sets, s1,s2,s3,s4,s5,s6. Here, s4,s5,s6 are the members of the tripartite graph and s1,s2,s3 are the immediate neighbours of the partite set, respectively. For this model, we have written an algorithm for finding the suitable labeling scheme according to the cardinality of s1,s2,s3. It will facilitate to execute the labeling function in an effective manner.Algorithm 1
Input: zero divisor graph Γ of a finite commutative ring of order η=pqr where p,q,r are distinct prime.Step 1: calculate the cardinality of the nontrivial divisor sets.∣s1∣=qr−q+r+1∣s2∣=pr−p+r+1∣s3∣=pq−p+q+1∣s4∣=p−1∣s5∣=q−1∣s6∣=r−1.Step 2: compare the cardinalities ofs1,s2, and s3 and find the greatest set.Step 3: if∣s1∣>∣s2∣ and ∣s1∣>∣s3∣Assign labeling scheme ass4⟶s5⟶s6⟶s1⟶s2⟶s3.Else if∣s2∣>∣s1∣ and ∣s2∣>∣s3∣Assign labeling scheme ass5⟶s4⟶s6⟶s2⟶s1⟶s3.ElseAssign labeling scheme ass6⟶s5⟶s4⟶s3⟶s2⟶s1.Output: determine the suitable labeling scheme of Γ.Theorem 5.
LetΓ be a zero divisor graph of order η=pqr where p,q,r are distinct prime. Then, λ2,1Γ=Δ+δ (maximum degree+minimum degree).Proof.
Event 1: if the labeling scheme iss4⟶s5⟶s6⟶s1⟶s2⟶s3.
(27)fvs4i=i−1,1≤i≤p−1,fvs5i=i+p−1,1≤i≤q−1,fvs6i=i+p+q−1,1≤i≤r−1,fvs1i=p+q−1ifi=1,i+p+q+r−3if2≤i≤qr−q+r+1.
The setss2 and s3 channels are at the distance of three from set s1, and they will get the same as the frequencies of s1.
Event 2: if the labeling scheme iss5⟶s4⟶s6⟶s2⟶s1⟶s2.
(28)fvs5i=i−1,1≤i≤q−1,fvs4i=i+q−1,1≤i≤p−1,fvs6i=i+p+q−1,1≤i≤r−1,fvs2i=p+q−1ifi=1,i+p+q+r−3if2≤i≤pr−p+r+1.
The setss1 and s3 channels are at the distance of three from set s2, and they will get the same frequencies of s2.
Event 3: if the labeling scheme iss6⟶s5⟶s4⟶s3⟶s2⟶s1.
(29)fvs6i=i−1,1≤i≤r−1,fvs5i=i+r−1,1≤i≤q−1,fvs4i=i+q+r−1,1≤i≤p−1,fvs3i=q+r−1ifi=1,i+p+q+r−3if2≤i≤pq−p+q−1.
The setss1 and s2 channels are at the distance of three from set s3, and they will get the same frequencies of s3. The above labeling executions in three different events as per their appropriate labeling scheme will yield λ2,1.Algorithm 2
Input: zero divisor graph Γ of a finite commutative ring of order η=pqr where p,q,r are distinct prime.Step 1: calculate the cardinality of the nontrivial divisor sets.∣s1∣=qr−q+r+1∣s2∣=pr−p+r+1∣s3∣=pq−p+q+1∣s4∣=p−1∣s5∣=q−1∣s6∣=r−1Step 2: compare the cardinalities ofs1,s2, and s3 and find the greatest set.Step 3: if∣s1∣>∣s2∣ and ∣s1∣>∣s3∣ OR ∣s2∣>∣s1∣ and ∣s2∣>∣s3∣Assign labeling scheme ass4⟶s5⟶s6⟶s1⟶s2⟶s3.ElseAssign labeling scheme ass6⟶s5⟶s4⟶s3⟶s2⟶s1.Output: determine the suitable labeling scheme of Γ.Theorem 6.
LetΓ be a zero divisor graph of a finite commutative ring of order η=pqr where p,q,r are distinct prime. Then, for a maximum degree Δ and minimum degree δ, λ3,2,1Γ=2Δ+δ+d+1, if the cardinality of the smallest prime and its multiple set is less than the sum of the cardinality of the other primes and its multiple sets where d is the difference of two cardinalities. Otherwise, λ3,2,1Γ=2Δ+δ.Proof.
Event 1: if the labeling scheme iss4⟶s5⟶s6⟶s1⟶s2⟶s3, then
(30)fvs4i=2i−2,1≤i≤p−1,fvs5i=2i+2p−3,1≤i≤q−1,fvs6i=2i+2p+q−4,1≤i≤r−1,fvs1i=2i+2p+q+r−6,1≤i≤qr−q+r+1ifs1>s2ands3,2i+2p+q+r−5,1≤i≤qr−q+r+1ifs2>s1ands3,fvs2i=2i+2p+q+r−5,1≤i≤pr−p+r+1ifs1>s2ands3,2i+2p+q+r−6,1≤i≤pr−p+r+1ifs2>s1ands3,fvs3i=2i+2q+2pr−3,1≤i≤pq−p+q+1ifs1>s2ands3,2i+2p+2qr−3,1≤i≤pq−p+q+1ifs2>s1ands3.
Event 2: if the labeling scheme iss6⟶s5⟶s4⟶s3⟶s2⟶s1, then
(31)fvs6i=2i−2,1≤i≤r−1,fvs5i=2i+2r−3,1≤i≤q−1,fvs4i=2i+2r+q−4,1≤i≤p−1,fvs3i=2i+2p+q+r−6,1≤i≤pq−P+q+1,fvs1i=2i+2p+q+r−5,1≤i≤qr−q+r+1,fvs2i=2i+2p+2qr−3,1≤i≤pr−p+r+1.
The execution of the above events will ensure theL3,2,1 number λ3,2,1Γ during the appropriate labeling scheme.Example 7.
Figure2 gives more interpretation on the L3,2,1 frequency assignment to the zero divisor graph Γ of a commutative ring of order η=7×11×13 whose ∣VΓ∣=280. This model comes under Theorem 6 and follows Event 1 labeling scheme by the instruction of Algorithm 2. The least upper frequency of Γ is 2Δ+δ+d+1=321 when executing appropriate labeling function on Γ.Figure 2
L3,2,1 labeling of Γ of order η=7×11×13.


## 3. Conclusion

In this paper, we handled some general classes of zero divisor graphs as frequency assignment networks. We have discussed the characteristics of their vertices and their edges. Algorithm construction for algebraic graphs is the most familiar, and it provides the systematic approach on these types of NP complete problems. Enormous applications are scattered as well behind every growing sciences; because of this reason, we took this graph under frequency labeling. Through this distance labeling, one can restrict a frequency domain along with findingλ of Γ in terms of Δ and δ. Our future research plan is to analyze some other classes of zero divisor graphs to find λ as well as break those networks by using the decycling technique which was introduced by Lowell W. Beineke and Robert C. Vandell in 1996.



---

*Source: 6698815-2021-03-09.xml*