Computing Edge Weights of Magic Labeling on Rooted Products of Graphs

Liu, Jia-Bao; Afzal, Hafiz Usman; Javaid, Muhammad

doi:https://doi.org/10.1155/2020/2160104

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Graph-Theoretic Techniques for the Study of Structures or Networks in Engineering

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Research Article | Open Access

Volume 2020 | Article ID 2160104 | https://doi.org/10.1155/2020/2160104

Computing Edge Weights of Magic Labeling on Rooted Products of Graphs

Jia-Bao Liu,^1,2Hafiz Usman Afzal,³and Muhammad Javaid³

Academic Editor: Alessandro Palmeri

Received10 May 2020

Revised05 Aug 2020

Accepted25 Aug 2020

Published29 Sept 2020

Abstract

Labeling of graphs with numbers is being explored nowadays due to its diverse range of applications in the fields of civil, software, electrical, and network engineering. For example, in network engineering, any systems interconnected in a network can be converted into a graph and specific numeric labels assigned to the converted graph under certain rules help us in the regulation of data traffic, connectivity, and bandwidth as well as in coding/decoding of signals. Especially, both antimagic and magic graphs serve as models for surveillance or security systems in urban planning. In 1998, Enomoto et al. introduced the notion of super edge-antimagic labeling of graphs. In this article, we shall compute super edge-antimagic labeling of the rooted product of and the complete bipartite graph combined with the union of path, copies of paths, and the star. We shall also compute a super edge-antimagic labeling of rooted product of with a special type of pancyclic graphs. The labeling provided here will also serve as super edge-antimagic labeling of the aforesaid graphs. All the structures discussed in this article are planar. Moreover, our findings have also been illustrated with examples and summarized in the form of a table and 3 plots.

1. Introduction

The antimagic and magic labelings on graphs are designed due to their wide applicability in various branches of engineering. In the literature, many results have appeared regarding numeric labelings on several operations of graphs such as graphs obtained from cartesian, corona, rooted, and strong products of various connected graphs; for instance, see [1–5]. In this article, we will provide super edge-antimagic labeling of the rooted product of and the complete bipartite graph taking its disjoint union with path, copies of paths, and star graph. Further, we will provide super edge-antimagic labeling of the rooted product of path with specifically designed pancyclic graphs. We shall, in particular, target the planar graphs that are obtained as a result of our rooted products. These planar graphs minimize the possibility of overlapping of various entities in practical purposes, which is a major cause of inefficiency in organizations. The super edge-antimagic labeling provided in this article on the specified graphs can be used as test-ready labeling in any engineering, networking, or industrial project where the scheme of design of connections is similar to the graphs obtained in this note.

1.1. Applications of Graph Labeling in Engineering

1.1.1. Software Engineering

In software engineering, the role of graph labeling is getting improved in the encryption of the security codes in order to halt the attacks of hackers on precious data and also in coding of data to transmit it to different networks and devices alike. Similarly, test-ready labels and reference labels are improving the configurations of software in designing the updated versions. A two-scan algorithm for the connected components in binary images involves labeling and is significantly helpful in making the graphics better and clearer (see [6]). In data mining, the concept of magic labeling is becoming useful with the passage of time. It makes the data collection for deriving new information easier by indicating the data of same weightage as one entity. In this way, magic labeling is making the task of data mining more easy, error-free, and less time- and effort-consuming in various organizations.

1.1.2. Networking

A network consists of nodes (vertices) and links between these nodes (edges). More technically, a network, say , is an ordered 2-tuple consisting of two sets, set of nodes and links between nodes called set of edges such that . Thus, a graph is directly a representative of a network. In network engineering, the optimization and functioning of the networks are the primary hallmarks that require solid planning, construction, and management of the network at its core. Two basic types of networking are wired networking and wireless networking. One cannot deny the existence, importance, and major usage of wired networking in many principal instances. But, due to more usefulness, the apparent increase in the use of the wireless networks demands the application of robust tools, like graph labeling, to get more accuracy in the engineering of wireless networking (see [7]). We are living in an era of communication, in which radio transmission is playing an extremely important part. These wireless radio networks face a major challenge in the form of interference which makes the task of channel assignment harder. The main reason for this unwanted interruption is constraint-free transmission of the concurrent networks admitting same instance appearance [8, 9]. Such networks are first converted into graphs and then magic labeling helps in assigning constant weights to the concurrent networks. This whole procedure minimizes and even eliminates the interference in networks. Moreover, the radio labeling of graphs is tremendously helpful in the minimization of the problem of interference in wireless networks and has been playing a very vital role in the last few years. The edge-antimagic labeling is particularly used for automatic routing in a network. A static network is represented first as a specific graph by connecting nodes in some topology to form a connected graph and then magic labeling is applied for automatic routing of data in the network. This labeling is designed with a constant edge weight, which helps routing to automatically detect the next node within that network (see [10]).

1.1.3. Telecommunication

The most commercially successful application of graph labeling appears in telecom engineering these days [11]. In telecommunication, a service coverage area is divided into a quadrilateral or a hexagon, termed as a cell in cellular networking. Furthermore, each cell works as a station. The base cell has the ability to communicate with mobile stations using its radio transceiver. The challenge for base cell here is to provide maximum channel reuse without violating the constraints in order to minimize the blocking. To tackle this challenge, a label is assigned to each user and the communication link of this user receives a distinct label. In this way, the numbers assigned to any two communicating terminals automatically specify the link label of the connecting path by simply using magic, antimagic, or graceful labeling. Conversely, the path label uniquely specifies the pair of users which it interconnects (see [12]).

1.1.4. Civil Engineering and Urban Planning

As a particular example, consider the wheel , helm , and prisms and in Figure 1. The edges of the graphs , , and are labeled with consecutive labels ranging from 1 up to the size of the graph such that the labels appearing on all the vertices are distinct. That is, the vertex antimagic labeling of these graphs is being provided here. Meanwhile, on , edge-magic labeling with constant edge weight 29 is provided [13, 14]. Now, for instance, in a surveillance design of highly sensitive office, the rooms can be represented by vertices and legitimate or specified passages to reach those rooms can be represented by edges. If someone tries to violate even a single legitimate passage, a complete disruption in the labeling will occur. As in case of structure like , the magic constant will get disturbed abruptly due to violation of passage. This will promptly indicate to the concerned security through a computer programme that someone has violated the passage and the security team will get alert. Now, these particular magic and antimagic schemes, once designed, can be used wherever they are needed in a similar security pattern. Both magic labeling and antimagic labeling are equally important in this regard. This is a major use of magic labeling and antimagic labeling in urban planning. Thus, these labelings play their part as model for surveillance or security for various types of buildings or areas [15].

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Moreover, the functioning and routing of robots installed in restaurants and in factories as production lines and other such units come under light using one of the labeling functions. As in robotics, they help to decide which function to be used and which to be skipped at a certain instance for making the robots or certain robotic components moving or keeping them static. The idea of robotic routing with the help of the tools like labeling functions and distance-based dimensions not only helps in minimizing the time for a robot for taking a decision but also maximizes its accuracy [16]. Such benefits are the cause of massive reduction of cost in industry.

We have organized this article into six main sections. Section 1 presents the introduction, followed by Section 2 of the preliminary definitions. Section 3 contains our main findings. Section 4 discusses the illustration of our findings through examples and proposed open problems. Section 5 focuses on the synopsis and 3 plots of our results. Conclusion is given in the last section.

2. Preliminary Definitions

In this section, we will discuss some definitions and results that are useful in the presentation of our theorems in the next sections. Also, salient works previously done in this field shall be mentioned here.

Let be a simple, connected, and nonempty graph with vertex set and edge set such that and , respectively. We call in this case a - graph. Throughout this article, we will use - graphs. For more insight into the graph theoretic terminologies, we refer the reader to [17].

A mapping that maps nonzero positive integers onto the vertices, edges, or both of a graph under certain conditions is called a labeling. It is called total labeling if we include both sets of vertices and edges in its domain. Some labelings carry the vertex set only or the edge set only in the domain and they are termed as vertex labelings or edge labelings, respectively. Two main types of labeling are magic labeling and antimagic labeling. In simple terms, magic labeling refers to equal vertex/edge weights and antimagic labeling refers to unequal vertex/edge weights.

Definition 1. For a - graph , the bijection is called edge-antimagic labeling (or edge-antimagic total labeling) if the edge weights , for each , form a sequence of consecutive positive integers with minimum edge weight and common difference . If such a labeling exists, then is said to be an edge-antimagic graph.

Definition 2. An edge-antimagic labeling is called a super edge-antimagic labeling of - graph if . Thus, a super edge-antimagic graph is one that admits a super edge-antimagic labeling.
In the above definitions, if we have , then the minimum edge weight becomes constant for all edges known as magic constant or magic sum of the graph .

Definition 3. A simple graph with is called a pancyclic graph if it contains cycle of every order from 3 to .

Definition 4. The rooted product of two graphs and is obtained by taking copies of and then, for every vertex of , identifying with the root node of the copy of . It is denoted by .
In 1963, Sadlácek defined the concept of magic labeling of graphs [18]. Later on, Hartsfield and Ringel [19] presented the idea of antimagic labeling for vertex-sums of a graph. The concept of edge-antimagic labeling of graphs was studied for the first time in [20] by Kotzig and Rosa who identified it by the name of magic valuation. In 1996, Ringel and Llado [21] studied this concept using a different terminology, that is, edge-antimagic labeling. Motivated by this concept, Enomoto et al., in 1998, defined the notion of super edge-antimagic labeling of graphs in [22]. They studied this concept with the term super edge-magic labeling of graphs. In the year 2000, Simanjantuk et al. introduced the idea of edge-antimagic labeling of graphs in [23].
The historical background of the edge-antimagic labeling of graphs includes the following important and interesting conjectures on trees.

Conjecture 1. Every tree is edge-antimagic [20].

Conjecture 2. Every tree is super edge-antimagic [22].

In support of Conjecture 2, many particular classes of trees have been studied by various authors. Lee and Shah [24] verified this conjecture for trees with at most 17 vertices with the help of a programming software. In particular, the results can be found for stars, subdivided stars [25–29], -trees [30–32], banana trees [33], caterpillars [34], subdivided caterpillars [35], and disjoint union of stars and books [36]. Further related studies can be seen in [37–39]. However, this conjecture is still open for working. In [22], it is proven that if a nontrivial - graph is super edge-antimagic, then . In the same article, the authors proved that a complete bipartite graph is super edge-antimagic if and only if or . In [40], it is proven that is super edge-antimagic if either is a multiple of or is a multiple of . Enomoto et al. [22] proved that is super edge-antimagic if and only if is odd. In [41], it has been proven that is super edge-antimagic if and only if and is even (also see [42]). In [43], Figueroa-Centeno et al. showed that the generalized prism is super edge-antimagic for every odd integer . In [3], Baig et al. presented the super edge-antimagic labeling of a class of pancyclic graphs. The following lemma regarding super edge-antimagic graphs is very useful.

Lemma 1 (see [43]). A -graph is super edge-antimagic total if and only if there exists a bijective function such that the set consists of consecutive integers. In such a case, extends to a super edge-antimagic total labeling of with magic constant , where and .

The sum in Lemma 1 is termed as edge sum for each edge . In fact, the set of all edge sums in Lemma 1 constituting an arithmetic progression forms a super edge-antimagic vertex labeling of the graph [23]. That is, constitutes a super edge-antimagic vertex labeling of . It means Lemma 1 states that whenever the set of edge sums forms a super edge-antimagic vertex labeling of , it extends to a super edge-antimagic labeling of .

We will frequently use the above lemma in the proofs of our results. Another very useful relevant result is as follows.

Theorem 1 (see [45]). If a -graph is super edge-antimagic, then it is super edge-antimagic always.

3. Main Results

This section consists of two further sections, in which we will present our main results. In Section 3.1, we will study the super edge-antimagic labeling of the disjoint union of the rooted product of and the complete bipartite graph with path, copies of paths, and the star. Meanwhile, in Section 3.2, we will provide the super edge-antimagic labeling of rooted product of with certain pancyclic graphs and . It is pertinent to mention here that all our graphs obtained as the result of the rooted products are planar.

3.1. Super Edge-Antimagic Labeling of the Disjoint Union of the Rooted Product of and with Path, Copies of Paths, and Star

The following open problem proposed by Ngurah et al. in [46] is our main motivation to study super edge-antimagic labeling of the graph containing copies of complete bipartite graph .

3.1.1. Open Problem

For and , can you determine any super edge-antimagic labeling of ?

The rooted product , in fact, contains copies of the complete bipartite graph . We swiftly move to our theorems now.

Theorem 2. For even and odd , the graph admits a super edge-antimagic labeling with magic constant .

Proof. Consider the graph with vertex and edge sets as follows:where and . We define a labeling as follows:The set of all edge sums generated by the labeling scheme forms a sequence of consecutive integers , which is a super edge-antimagic vertex labeling of . Therefore, by Lemma 1, extends to a super edge-antimagic labeling of the graph with magic constant .

Theorem 3. For odd , the graph admits a super edge-antimagic labeling with magic constant .

Proof. Consider the graph with odd as follows:where and . We define a labeling as follows:The set of all edge sums generated by the above labeling scheme forms a sequence of consecutive integers , which is a super edge-antimagic vertex labeling of . Therefore, by Lemma 1, extends to a super edge-antimagic labeling of the graph with magic constant .

Theorem 4. For even and odd , the graph admits a super edge-antimagic labeling with magic constant .

Proof. Consider the graph , for odd , constructed aswhere we have and . Now, we define a labeling as follows:The set of all edge sums generated by the above labeling scheme constitutes a sequence of consecutive integers , which is a super edge-antimagic vertex labeling of . Therefore, by Lemma 1, extends to a super edge-antimagic labeling of the graph with magic constant .

Theorem 5. For odd , the graph admits a super edge-antimagic labeling with magic constant .

Proof. Consider the graph , for odd , with the construction:where and . Now, consider a labeling as follows:The set of all edge sums generated by the above labeling scheme constitutes a sequence of consecutive integers , which is a super edge-antimagic vertex labeling of . Therefore, by Lemma 1, extends to a super edge-antimagic labeling of the graph with magic constant .

Theorem 6. For even and odd , the graph admits a super edge-antimagic labeling with magic constant .

Proof. Consider the graph , for odd , with the construction:where we have and . We define a labeling as follows:The set of all edge sums generated by the above labeling scheme forms a sequence of consecutive integers , which is a super edge-antimagic vertex labeling of . Therefore, by Lemma 1, extends to a super edge-antimagic labeling of the graph with magic constant .

Theorem 7. For odd , the graph admits a super edge-antimagic labeling with magic constant .

Proof. Consider the graph for both odd and with vertex and edge sets as follows.where and . Now, we define a labeling as follows:The set of all edge sums generated by the above labeling scheme forms a sequence of consecutive integers , which is a super edge-antimagic vertex labeling of . Therefore, by Lemma 1, extends to a super edge-antimagic labeling of the graph with magic constant .

3.1.2. Observations

(1)In Theorems 2, 4, and 6, the labels for & will receive the values only when . Similarly, in Theorems 3, 5, and 7, the labels for will receive the values when . This happens because and has 3 vertices in one partitioned set. However, the labeling of our graphs never gets disturbed anyway (here, ).(2)The following results from Theorems 2–7 are direct consequences of Theorem 1.(3)As a cycle is super edge-antimagic if and only if is odd [22], this means that is not super edge-antimagic. We have observed an interesting substructure in Theorems 2, 4, and 6. That is, if we fix in these theorems, we obtain a cyclic family of graphs, in which only cycle appears times ( is odd). Obviously, these families are also super edge-antimagic, as per the proofs of our results. See super edge-antimagic labeling of , , and in Figure 2.

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Theorem 8. For all and odd , the graph admits a super edge-antimagic labeling.

Theorem 9. For all and odd , the graph admits a super edge-antimagic labeling.

Theorem 10. For all and odd , the graph admits a super edge-antimagic labeling.

3.2. Super Edge-Antimagic Labeling of Rooted Product of with Certain Pancyclic Graphs

In networking, the networks that are acyclic and networks containing a range of cycles admit same level of importance. In graph theoretic terms, the former corresponds to trees and the latter corresponds to multicyclic graphs. What about a network that contains cyclic connection of all orders starting from 3 up to the number of systems connected within that network? In such situations, the task of the programmers gets more tough, as they have to work on encrypting powerful codes to keep their data safe by halting the attacks of hackers. It is because every system in such network is connected within a cycle. This kind of network corresponds to a pancyclic graph (see Definition 3).

We define here a specific pancyclic graph as follows.

Definition 5. Consider a pancyclic graph with the construction:In the next result, we will show that the rooted product is super edge-antimagic for odd values of .

Theorem 11. For odd , the rooted product admits a super edge-antimagic labeling with magic constant .

Proof. (1)For , . The vertex labeling extends to a super edge-antimagic labeling of by Lemma 1.(2)For , consider the graph with and consisting of the following vertex and edge sets:Consider a labeling defined asThe set of all edge sums generated by the above labeling scheme forms a sequence of consecutive integers , which is a super edge-antimagic vertex labeling of . Therefore, by Lemma 1, extends to a super edge-antimagic labeling of the graph with magic constant .

Definition 6. Consider a pancyclic graph (nonisomorphic to ) with the vertex-edge connection as follows:

Theorem 12. For odd , the rooted product admits a super edge-antimagic labeling with magic constant .

Proof. Consider the rooted product with and with vertex and edge sets as follows:And the labeling scheme is the same as designed in Theorem 11.
The following result is a direct consequence of Theorem 1.

Theorem 13. For odd , the rooted products and are super edge-antimagic.

4. Illustration through Examples and Proposed Open Problems

4.1. Examples

As examples of Theorems 2 and 3, the super edge-antimagic labeling of and super edge-antimagic labeling of are presented in Figures 3 and 4, respectively. It is evident that the magic constant of the labeling in Figure 3 is 132 (corresponding to the parameters and ) and the magic constant of the labeling in Figure 4 is 147 (corresponding to the parameters and ). These are as per the magic constants depicted in the proofs of Theorems 2 and 3.

Similarly, Figures 5–8 illustrate Theorems 4–7, respectively, for fixed parameters mentioned in the caption of each. Here, the magic constants are as per our depiction as well.

Figures 9 and 10 illustrate super edge-antimagic labeling of the rooted products appearing in Theorems 11 and 12 with parameter .(1)In all of the above figures, Lemma 1 has helped us in the way that we do not require to label the edges. We only have observed that the edge sums are consecutive positive integers. So, applying the remaining labels to the edges of the graph in descending or ascending sequence of edge sums will give rise to super or edge-antimagic labeling of the graph, respectively, for appropriate values of and .

4.2. Open Problems

Here, we propose few open problems related to the results (Theorems 2–7) presented in Section 3.1 as follows:(1)For even , can you determine any super edge-antimagic labeling of ?(2)For even , can you determine any super edge-antimagic labeling of ?(3)For even , can you determine any super edge-antimagic labeling of ?(4)For odd , can you determine super edge-antimagic labeling of the graphs , , , , , and for a magic constant (i.e., for any other value of ) other than those obtained in this article?(5)For , find any super edge-antimagic labeling of the following graphs:(1)(2)(3)

The open problems related to the results (Theorems 11 and 12) presented in Section 3.2 are as follows:(1)For even , can you determine any super edge-antimagic labeling of the rooted product ?(2)For even , can you determine any super edge-antimagic labeling of the rooted product ?(3)For odd , can you determine any super edge-antimagic labeling of the rooted products and with a magic constant that is different from those obtained here, that is, for any other value of

5. Synopsis and 3 Graphical Trends and Relative Comparison of the Magic Constants

5.1. Synopsis

The computational results of our findings have been summarized in Table 1. The “Parameters” column represents those parameters for which we have been able to find super and edge-antimagic labeling of the corresponding graph.

5.2. 3 Graphical Comparison

The following are the 3 plot representations of the magic constants obtained in our results against the parameters and , shown in Figure 11 ( represents magic constant in the plots). Meanwhile, Figure 12 shows the comparison of these magic constants against the said values of the parameters. Among these magic constants, the most dominant magic constant, especially in the comparison of , , and , is of the graph (as red layer is the dominant one). It means that, among the magic constants of the networks discussed here, the disjoint union of the rooted product and is having the highest values.

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6. Conclusion

In this note,(1)we have obtained a super edge-antimagic labeling of the rooted product of with the complete bipartite graph taking its union with path, copies of paths, and star for possible values of . The obtained results partially point towards the problem proposed in [46] regarding ;(2)we have presented a super edge-antimagic labeling of the rooted product of with special pancyclic graphs and ;(3)importantly, all the resultant graphs of the rooted products here are planar graphs;(4)several open problems have also been proposed in this article for further working in this area;(5)The obtained schemes are now all set to serve as test-ready labelings for engineers, programmers, and networking professionals to utilize them where they find them suitable.

Data Availability

The whole data are included within this article. However, more details on the data can be obtained from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work was supported in part by the China Postdoctoral Science Foundation under Grant 2017M621579, in part by the Postdoctoral Science Foundation of Jiangsu Province under Grant 1701081B, and in part by the Project of Anhui Jianzhu University under Grant 2016QD116 and Grant 2017dc03.

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Copyright © 2020 Jia-Bao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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