Abstract
Given a frequency assignment network model is a zero divisor graph of commutative ring , 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 and labeling constraints. For a graph , labeling is a nonnegative real valued function such that if and if where and are any two vertices in and is a distance between and . 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 labeling where 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 and , within , in terms of maximum and minimum degree of by the distance labeling and , respectively, for some order where are distinct prime and is any positive integer.
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 as . 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 analogue was first proposed by Griggs and Yeh in 1992 [4], and they discussed labeling bound for some basic graphs in terms of maximum degree . According to Griggs, for a graph , an labeling is a nonnegative real valued function such that if and if where and are any two vertices in and is the distance between and . The labeling number of is the smallest number such that no other label does not exceed that and is denoted by . The canonical generalization of labeling is labeling where is the difference of any two vertex frequencies connected by a two length path and is the minimum spanning labeling number.
At this instance, our FAP input model is zero divisor graph of a commutative ring [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 be a set of all nonzero zero divisors of , which is the vertex set . An edge between two distinct vertices and if and only if . Our aim of output is to attain minimum spanning frequency such that no other frequency is greater than under the constraint of distance. Suppose, for a (frequency model) graph, has a diameter two then labeling (assignment) as same as labeling (assignment) of . Let be a frequency function and be a distance of from , on the vertex set to the frequency domain where be a frequency for any vertex (channel) . Suppose , then is a labeling function of if for all
Suppose , then is a labeling function of if for all
Note that our FAP follows the static model [1] FCA, and then, the connections remain the same ever. For a graph , the general upper bound on is never much larger than said by Griggs [4] and also he proved . Over time, this general bound was again restricted as by Chang and Kuo [6]. In this manuscript, we determined the sharp upper bound on and of a zero divisor graph of a commutative ring, in terms of (maximum degree of ) and (minimum degree of ) 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 .
2. Upper Bounds in terms of and
In this section, we have obtained the exact bound on and labeling number for some classes of zero divisor graph of finite commutative ring whose order is denoted by . Throughout this paper, we used as a nonnegative real valued function for labeling of for some order is and . It is very essential to analyze the structure of the graph, and it is clear that . These zero divisors are contributed by each nontrivial divisor of . Let and be the nontrivial divisors of the ring of order . Consider the set , where is the unique multiple of and is the contribution of vertices (zero divisors) by nontrivial divisors of , including itself such that, for any
Therefore, . Now, the problem is to climb the cardinality of by inductive steps
By observation, we get
In particular, From equation (6), we can construct the set and . Let be a set of all vertices of such that , where is the number of unique multiple of , and by the previous discussion, for any , .
Hence,
When the summation is taken from , we get
From equations (9), (8), (6), and (5), one can find the cardinality of vertices contributed by the nontrivial divisors of , which are mentioned initially. By adding those equations, we get which is the number of zero divisors of a ring as well as the vertex set ; in addition, the author will express the maximum degree and minimum degree of in terms of and such that of for a ring with order where and are distinct prime and is any positive integer.
Theorem 1. Let be a zero divisor graph of order where and are distinct prime and is any positive integer. Then, where and are maximum and minimum degrees of , respectively.
Proof. Type 1. Suppose .
Case 1. If , then the labeling scheme .
Here, ’s denote the vertices of complete subblock whereas ’s denote the other vertices of .
Case 2. If , then the labeling scheme .
As per requirement, we can assign the distinct labels for all from the same labeling domain of . Since in addition the distance from to and to , vertices of are three.
Subcase 1. If , then such graphs have no ’s in .
In this subcase, the set vertices are at the distance of three from . The analogue of this labeling allows to assign the same labeling of to the vertices of .
Type 2. Suppose .
Case 1. If , then the labeling scheme elements.
At this instance, vertices of can be labeled by the numbers between the two existing labels that are assigned to the vertices of Case 2. If , then the labeling scheme .
At the end of this execution, there are missed numbers between the existing labels of which can be accessed and assigned as the labels, for some , with the next step of scheme execution as follows:
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. Let be a zero divisor graph of a ring of order where and are distinct prime and is any positive integer. Then, for a maximum degree of ,
Proof. Since is a zero divisor graph of a finite commutative ring of order , the set of all nonzero zero divisors is the vertex set of cardinality . Any two vertices and of are said to be adjacent if to where zero is the additive identity. The vertex set has the partition such that , which are the unique multiples of , , respectively. An element belongs to any one of the nontrivial divisor collection such that ; this will ensure the unique existence of the element in each set. An edge belongs to the different partition or in the same partition; the corresponding vertices are zero to at least prime factor of . In , multiples of nontrivial divisor collections are adjacent to all other element of the same collection and they individually form the complete graphs; at the same time, they together form in , which is denoted by . The remaining elements are not adjacent to one another, which is denoted by .
Type 1. Suppose , then the labeling scheme .
It is clear that the minimum distance from is three to and elements of . For assigning labels to the elements of , at this instance, we can expect two possibilities and .
If , then
If , then
Type 2. Suppose , then
Case 1. If , then
Case 2. If , then
Thus, we have attained the number of which is minimum.
Example 3. Figure 1 describes the FAP model which resembles a zero divisor graph of a finite commutative ring of order along with their frequency assignments. For this ring, the cardinality of zero divisors acts as a number of channels (vertices) such that . The vertices are partitioned into four separate blocks, namely, , , , and , respectively. Note that , , , and . The dotted subblock and the darkened subblock of together form the complete graph which is denoted by where the dotted subblock alone is adjacent with block . The other subblock of is adjacent with both darkened and dotted subblock but not within itself. In , the dotted subblock alone is adjacent with block. Adjacency relation between and 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 which is where .
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 graph of a finite commutative ring of order , which is isomorphic to the split graph. Let be a set consisting the vertices of the complete graph and the collection of independent vertices is said to be set . Both sets are having the cardinalities and , respectively. It is clear that are the set of zero divisors of ring . Therefore, the maximum degree is and the minimum degree is .
Theorem 4. Let be a zero divisor graph of order where is prime and is any positive integer. Then, and where and denote the maximum and minimum degrees of , respectively.
Proof. For this model, we discussed two different frequency distributions. Let be a frequency distribution function which is executed in two distributions with the same labeling scheme but different distance constraints.
Frequency Assignment 1
Frequency Assignment 2
By applying the labeling function in each assignment, we can obtain the number and the number of , 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 order where are distinct prime. For that, are the nontrivial divisors of the graph. Each divisor and its multiples together form a collection of vertices of that are indexed as the sets, . Here, are the members of the tripartite graph and 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 . It will facilitate to execute the labeling function in an effective manner.
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Theorem 5. Let be a zero divisor graph of order where are distinct prime. Then, (maximum degree+minimum degree).
Proof. Event 1: if the labeling scheme is .
The sets and channels are at the distance of three from set , and they will get the same as the frequencies of .
Event 2: if the labeling scheme is .
The sets and channels are at the distance of three from set , and they will get the same frequencies of .
Event 3: if the labeling scheme is .
The sets and channels are at the distance of three from set , and they will get the same frequencies of . The above labeling executions in three different events as per their appropriate labeling scheme will yield .
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Theorem 6. Let be a zero divisor graph of a finite commutative ring of order where are distinct prime. Then, for a maximum degree and minimum degree , , 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 is the difference of two cardinalities. Otherwise, .
Proof. Event 1: if the labeling scheme is , then Event 2: if the labeling scheme is , then The execution of the above events will ensure the number during the appropriate labeling scheme.
Example 7. Figure 2 gives more interpretation on the frequency assignment to the zero divisor graph of a commutative ring of order whose . This model comes under Theorem 6 and follows Event 1 labeling scheme by the instruction of Algorithm 2. The least upper frequency of is when executing appropriate labeling function on .
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 finding of in terms of and . Our future research plan is to analyze some other classes of zero divisor graphs to find as well as break those networks by using the decycling technique which was introduced by Lowell W. Beineke and Robert C. Vandell in 1996.
Data Availability
The nature of the problem is to find the minimum sharp upper numerical bound for some general classes of zero divisor graph. This bound was obtained by and labeling techniques. This is popularly known as channel assignment problem. The data that support the findings of this study are available in the research article, about the graph [https://scholar.rose-hulman.edu/rhumj/vol17/iss2/6 and https://rb.gy/oa3nbs] and labeling [https://rb.gy/nvywej, DOI: 10.1016/j.aml.2006.08.029, 10.1016/j.endm.2017.11.013]. In 1992, Griggs conjectured that for any graph, frequency upper bound $\lambda \leq \Delta^2+2\Delta$. This was refined to $\Delta^2+\Delta$ by Chang, Kuo, and Havet et al. [DOI: 10.7151/dmgt.2086]. But we have claimed this frequency bound for general classes of zero divisor graphs with in $\Delta^2$, where $\Delta$ is the maximum degree of the graph.
Conflicts of Interest
No potential conflict of interest was reported by the authors.