Computing Edge Weights of Symmetric Classes of Networks

Afzal, Hafiz Usman; Javaid, Muhammad; Alanazi, Abdulaziz Mohammed; Alshehri, Maryam Gharamah

doi:https://doi.org/10.1155/2021/5562544

Mathematical Problems in Engineering

On this page

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

Special Issue

Graph Invariants and Their Applications

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 5562544 | https://doi.org/10.1155/2021/5562544

Computing Edge Weights of Symmetric Classes of Networks

Hafiz Usman Afzal,¹Muhammad Javaid,¹Abdulaziz Mohammed Alanazi,²and Maryam Gharamah Alshehri²

Academic Editor: Ali Ahmad

Received22 Jan 2021

Revised17 Feb 2021

Accepted01 Mar 2021

Published20 Mar 2021

Abstract

Accessibility, robustness, and connectivity are the salient structural properties of networks. The labelling of networks with numeric numbers using the parameters of edge or vertex weights plays an eminent role in the study of the aforesaid properties. The systems interlinked in a network are transformed into a graphical network, and specific numeric labels assigned to the converted network under certain rules assist us in the regulation of data traffic, bandwidth, and coding/decoding of signals. Two major classes of such network labellings are magic and antimagic. The notion of super edge-antimagic labelling on networks was identified in the late nineties. The present article addresses super edge antimagicness of union of the networks’ star , the path , and copies of paths and the rooted product of cycle with . We also provide super edge-antimagic labelling of the rooted product of cycle and planar pancyclic networks. Further, we design a super edge-antimagic labelling on a pancyclic network containing chains of and three different symmetrically designed lattices. Moreover, our findings have also been recapitulated in the shape of 3- plots and tables.

1. Introduction

In this section, we shall define our problem and explain the objective of this study in Section 1.1, followed by Section 1.2, consisting of the definitions and results which we will use in our findings. Some previously performed work in this area will also be discussed in this section. Moreover, Section 1.3 concerns with applications of antimagic and magic labelling in various branches of networking, engineering, and computer science.

1.1. Problem Definition and Objective of the Study

In the fields of networking and computer science, the magic and antimagic labelling on networks are designed due to their extensive applications. Numerous results have been obtained on numeric labelling of several operations on networks such as Cartesian, lexicographical, corona, and modular products of various kinds of connected networks (see [1,2] for instance). The present article addresses super edge-antimagic labelling of the rooted product of and taking its disjoint union with the star, path, and copies of paths. We shall also design super edge-antimagic labelling on rooted product of specifically designed planar pancyclic networks with cycle . Moreover, we shall design super edge-antimagic labelling on planar pancyclic networks containing chains of and three different symmetric lattice networks (notated as , , and ). Except lattice networks, interestingly, all networks discussed in this note are planar. The overlapping probability of various networking elements minimizes in the course of planar networks. In organizations, this issue of entities’ overlapping is one of the major reasons of inefficiency. The test ready antimagic labellings obtained in this note on particular networks can be utilized in various projects of computer science and engineering admitting suitable and equivalently designed schemes of networking.

1.2. Definitions and Preliminaries

Some useful definitions and preliminary results in the context of this note shall be discussed in this section. We will also mention some relevant studies previously done in this field.

An ordered 2-tuple comprising two sets, i.e., set of nodes termed as vertex set and connection between these vertices termed as edge set , is called a network , where is contained in . A network can either be connected or consists of connected components. We will consider nonempty and simple networks throughout, having as its vertex set and as its edge set with order and size . The network in this case is termed as a -network. Reference [3] is referred to for further discernment into the network terminologies.

A labelling is a function that maps + integers (nonzero) onto the component(s) of a network under specified constraints. If the components include vertices and edges both, then this labelling is termed as total. The labellings are referred to as vertex or edge labellings if they cover, respectively, or alone in the domain. Magic and antimagic labelling are two main classes of labelling. Precisely, equal or unequal vertex/edge weights refer to magic or antimagic labellings, respectively.

Definition 1. For a-network, the bijective functionformontois termed asedge-antimagic total labelling with the constraint that the edge weights, for each, constitute a sequence of consecutive positive integers, whereis the minimum edge weight and common difference is.is referred to be anedge-antimagic total network, if such a labelling exists.

Definition 2. A superedge-antimagic total labelling is in fact anedge-antimagic total labelling in which minimum labelsare assigned to vertices of the-network.is termed to be a superedge-antimagic total network in this case.
In Definitions 1 and 2, the minimum edge weight becomes constant at , for all edges , which is referred to as magic sum or magic constant for the network .

Definition 3. A networkis termed as a pancyclic network if it contains cycle of every order from 3 to.

Definition 4. Letandbe two simple networks. The network obtained by taking copies ofandthen for each point (vertex) in V (called the root vertex) (), is being replaced with the copy of, termed as the rooted product of the networks and. It is notated as.
Further in the article, the abbreviations are being used as given in. Table 1.
We further provide some specific definitions within the corresponding section of our main results’ section.
The idea of magic labelling on networks was identified by Sadlácek in 1963 [4]. The notion of antimagic labelling, for vertex sums of networks, was presented by Ringel and Hartsfield [5] later. Kotzig and Rosa brought into the light the concept of magic valuations of networks in [6] which was in fact the -EAM total labelling on networks (studied by Ringel and Llado [7] in 1996). The notion of S--EAM total labelling of networks was defined by Enomoto et al. [8] with the terminology super edge-magic labelling. Simanjantuk et al. highlighted -EAM total labelling of networks in [9] in year 2000.
The literature of -EAM total labelling of networks includes the following interesting and useful conjectures.

Conjecture 1. All trees admit (a, 0)-EAM total labelling [6].

Conjecture 2. All trees admit S-(a, 0)-EAM total labelling [8].

Graph theorists, in the support of Conjecture 2, have been rectifying several particular classes of trees. Using an encryption of a computer programme, this conjecture has been verified for the trees having at most 17 vertices by Lee and Shah [10]. Specifically, the derivations can be seen for stars, subdivided stars [11, 12, 13, 14, 15], -trees [16, 17, 18], banana trees [19], caterpillars [20], subdivided caterpillars [21], and the union of books and stars [22]. Further relevant works can be found in [23–25]. However, this conjecture is still open for working. Enomoto et al. proved that if a simple -network is S--EAM total, then is at least [8]. They further derived that the network is S--EAM total or is 1. Figueroa-Centeno et al. derived that the union of networks is S--EAM total if either or [26]. The network is also proved to be S--EAM total only if in [8]. In [27], has been proven to be S--EAM total only when . The generalized prism is proven to be S--EAM total for all odd values of in [28]. Baig et al. classified a class of planar pancyclic networks in [29] and exhibited its S--EAM total labelling for all possible values of the parameters involved. An immensely advantageous lemma on S--EAM total networks is as follows. Liu at al. studied the bounds of the minimum and maximum edge weights for super - EAM total labelling on a generalized class of subdivided caterpillars in [30] for various values of . In [31], Ahmad et al. studied the super -EAM total labelling of certain Toeplitz graphs combined with isolated vertices , for various values of (also known as super edge-magic deficiency of networks). The properties and existence of super vertex-antimagic labelling of regular graphs have been discussed in [32]. In [33], Ahmad et al. constructed the - labelling, a special case of graceful labelling (labelling in which distinct edge weights are considered with respect to the difference of vertices’ labels) on trees, and transformed this labelling to edge-antimagic vertex labelling of trees. In [34], S--EAM total labelling on the graphs has been studied, where represents the unicyclic graph, whereas S--EAM total labelling of networks like zig-zag triangle and disjoint union of combs and stars has been studied in [35].

Lemma 1 (see [28]). A-networkis S--EAM total if and only if there is a bijectionsuch that the setconsists ofconsecutive integers. In such a case,extends to an S--EAM total labelling ofwith magic constant, whereand.

In Lemma 1, the sum is called as edge sum for each edge . This lemma will be used frequently in our derivations, as it keeps this sufficient to label the vertices of a network only to make the network S--EAM total, if the edge sums are positive consecutive integers. The following result is also very pertinent as far as S--EAM total networks are concerned.

Theorem 1. A simple network G admits an S-(a, 0)-EAM total labellingG admits an S-(a-+ 1, 2)-EAM total labelling [36].

1.3. Applications in Networking, Computer Science, and Engineering

In software engineering, network labelling keeps on attaining an improved role in the security codes’ encryption in order to encounter the attacks of trojans onto the precious data designed by hackers and also in designing of algorithm that helps the transmission of data to various networks and similar devices. The configurations of software in the encryption of their updated version is being improved by the use of reference labels and test ready labels nowadays. For connected components of networks in binary graphics, the mechanism which is predominantly nurturing the creation of clearer graphics involves labelling [37]. The study of magic labelling has been appearing to be more useful gradually in the data mining. The task of collection of data for the derivation of latest information gets more uncomplicated by designating equal weightage data as a single element. Resultantly, in organizations, the data mining task is becoming facile and more simplistic with far less consumption of time and effort due to the usage of magic labelling.

1.3.1. Networking

The primary hallmarks in networking are the functioning and optimization of the networks that demand management, construction, and concrete planning of networks at its base. Wireless and wired networking are two fundamental types of networking. The importance and large-scale usage of wired networking cannot be denied in the present era as well. The application of robust tools like network labelling is getting attention due to an escalation in the usage of wireless networking, in order to attain more precision in this field (see [38]). The modern era is of network communication whose part and parcel is radio transmission. The interference, making the job of channel assignment more complicated, is one of the major concerns in radio transmission. The transmission of concurrent networks that are constraint-free, admitting same instance surfacing, is the central reason of this unwanted interruption [39,40]. The magic labelling assists in the allotment of constant weights to the networks that are concurrent. Such interferences are eliminated by using this procedure. The radio labelling on networks is playing a tremendous part in the reduction of interference issue in wireless networking from the last decade or so. For the automatic routing in networks, the -antimagic labelling is particularly very useful. In this regard, a suitable constant edge-weight function is designed on a particular network, which helps routing for automatic detection of the succeeding node in the network (see [41]).

1.3.2. Telecommunication

In modern era, telecommunication involves most successful application of network labelling commercially [42]. In network telecommunication, a utility coverage region is split into a polygonal area described as a cell. Such a cell serves as a separate station. Using its radio transceiver, the base cell is designed to be a hub with the capacity to interface with other mobile stations. The defiance task here for the base cell is to facilitate with the ability to re-use utmost channels, avoiding any violation of the constraints. This challenge is being tackled by assigning a label to each user, while the communication loop of this user acquires a distinct label. Resultantly, any pair of communication terminals identifies the link label of connection path automatically by simple use of graceful antimagic or magic labelling. The label of the path specifies uniquely the two users which it interlinks conversely (see [43]).

1.3.3. Urban Planning

Consider the wheel , the helm , and prism in Figure 1 as a specific example. The edges of the networks and are labelled with consecutive labels ranging from 1 up to the size of the network such that the label appearing on all the vertices is distinct, i.e., we are provided here with the vertex antimagic labelling of the networks, whereas with edge weight 29 (constant), edge-magic labelling on is given [44,45]. As an example, the chambers are identified by point (vertices) and admissible pathways to approach these chambers are identified by edges, in a surveillance design of highly secured building. A total disturbance in the labelling will occur if a person attempts to breach a single legal pathway. The magic constant, in the scenario of design like , gets disordered promptly in case of violation in the pathway. This disorder, through programming software, will abruptly alert to the security concerned that the legitimate pathway has been breached. Once such magic or antimagic labellings are designed on a network, they can be used for surveillance of all the networks having the same hubs and connections. Antimagic and magic labelling both are equally valuable in this regard. In urban planning, this is one of the large-scale usages of the concept of labelling. That is, as a model for surveillance of the extensively secure areas, these labellings perform their distinct role [46].

(a)

(b)

(c)

1.3.4. Robotics

The routing and functioning of inducted robots at places like restaurants and factories in the form of production lines and machine units derive assistance by making use of any such suitable labelling function. In order to keep robotic components kinetic or make them stationary, these labelling functions assist to opt which operation to be skipped at which instant and vice versa. The antimagic labelling and distance-based dimensions alike tools help to minimize the time and maximize the accuracy of robots in their routing [47]. In the industry, these tools are causing a massive reduction in the cost.

2. Main Results

This section contains our main findings. It is divided into four subsections further. In Section 2.1, the S--EAM total labelling on the union of with copies of paths, the star, and the path shall be designed, whereas in Section 2.2, we derive an S--EAM total labelling on the rooted product of network and pancyclic networks and (planar also). Further, S--EAM total labelling of planar pancyclic network and symmetric lattice networks , , and shall be exhibited in Sections 2.3 and 2.4, respectively.

2.1. S--EAM Total Labelling of the Disjoint Union of and , , and

Our main motivation to explore the findings in this section is the following open problem of Ngurah et al. [48].

Open Problem. For and , is there any S--EAM total labelling of ?

In fact, the rooted product contains copies of the complete bipartite network . A cycle is a 2-regular network of order , whereas the complete bipartite network is class-wise regular in which one partitioned class of vertices is 2-regular and other is -regular, where .

Theorem 2. For and , the networkadmits an S--EAM total labelling having.

Proof. Consider the network with connection scheme as below:Here, and . Define a labelling as(i)For :(ii)For :(iii)The remaining labels for are as follows:All edge sums generated by the labelling scheme constitute a sequence of consecutive integer . So, by Lemma 1, extends to an S--EAM total labelling of the network having magic constant .

Theorem 3. For and , the networkadmits an S--EAM total labelling with.

Proof. Consider , a network with odd as follows:Here, and . We define a labelling as follows:(i)For :(ii)For :(iii)The remaining labels for are as follows:All edge sums generated by the above labelling scheme constitute a sequence of consecutive integer . By Lemma 1, extends to an S--EAM total labelling of the network having .

Theorem 4. For and , the networkadmits an S--EAM total labelling with.

Proof. Let be a network for with vertices:Here, and . A labelling function is defined as follows:(i)For :(ii)For :(iii)The remaining labels for are as follows:All edge sums generated by the above labelling scheme constitute a sequence of consecutive integer . So, by Lemma 1, extends to an S--EAM total labelling of the network admitting magic constant .

Theorem 5. For and , the networkadmits an S--EAM total labelling with.

Proof. Consider the network , for odd with the construction:Here, and . Define a function as follows:(i)For :(ii)For :(iii)The remaining labels for are as follows:All edge sums generated by the above labelling scheme constitute a sequence of consecutive integer . So, by Lemma 1, extends to an S--EAM total labelling of the network admitting .

Theorem 6. For and , the networkadmits an S--EAM total labelling with.

Proof. Consider the network , for odd, with connections:Here, and . A labelling is defined as follows:(i)For :(ii)For :(iii)The remaining labels for are as follows:All edge sums generated by the above labelling scheme constitute a sequence of consecutive integer . Therefore, by Lemma 1, extends to an S--EAM total labelling of the network admitting magic constant .

Theorem 7. For and , the networkadmits an S--EAM total labelling with.

Proof. Consider the network , for both and , as follows:Here, and . A labelling is designed as follows:(i)For :(ii)For :(iii)The remaining labels for are as follows:The edge sums generated by the above labelling scheme constitute a sequence of consecutive integer . By Lemma 1, extends to an S--EAM total labelling of the network having magic constant .
Observations. The network is S--EAM total for only [8], pointing out that is not S--EAM total. The networks in Theorems 2, 4, and 6 contain interesting substructures. Keeping fixed for these results, S--EAM total families of networks involving copies of can be obtained. For instance, S--EAM total labelling (again by using Lemma 1) of , , and is presented in Figure 2.
The following results from Theorems 2–7 are direct consequences of Theorem 1.

(a)

(b)

(c)

Theorem 8. and ,admits an S--EAM total labelling.

Theorem 9. and ,admits an S--EAM total labelling.

Theorem 10. and ,admits an S--EAM total labelling.

2.2. S--EAM Total Labelling of Rooted Product of Pancyclic Networks with

The present section deals with S--EAM total labelling of the rooted product of two specific planar non-isomorphic pancyclic networks and the cycle .

A specific pancyclic network is defined as follows.

Definition 5. is a pancyclic network having the following construction.

Theorem 11. For, the rooted productadmits an S--EAM total labelling having magic constant.

Proof. (i)For , . The vertex labelling extends to an S--EAM total labelling of , by Lemma 1.(ii)For . Consider the network with and connected as per the following scheme:Consider a labelling defined asAll edge sums generated by the above labelling scheme form a sequence of consecutive integer . Therefore, by Lemma 1, extends to an S--EAM total labelling of the network having magic constant .

Definition 6. We define a pancyclic networkhaving vertex set:

Theorem 12. For, the rooted productadmits an S--EAM total labelling with magic constant.

Proof. (i)For , .(ii)For .Consider with and with the following connection:The labelling scheme for and is the same as designed in Theorem 11.
A direct derivation from Theorem 1 is given as follows.

Theorem 13. For,andare S--EAM total.

2.3. S--EAM Total Labelling of a Pancyclic Class of Networks: Extension of a Result Appearing in [29]

In [29], Baig et al. provided a result regarding S--EAM total labelling of a pancyclic class of networks involving chains of cycle . Here, we shall introduce a pancyclic family of networks involving chains of cycle , while our point of convergence is the S--EAM total labelling of this class. Thus, we further extend the results of Baig et al. [29].

Definition 7. The pancyclic networkis a network with orderand, with structure as follows:Figure 3 reveals general formation of .
In Figure 4, we have shown the network and its contained cycles of orders .
In the upcoming result, we show that the pancyclic network is S--EAM total.

Theorem 14. For all positive integers, the pancyclic networkis S--EAM total having magic constant.

Proof. (i)For , the labelling extends to an S--EAM total labelling of by Lemma 1.(ii)For .Define here a labelling asAll edge sums generated by the above labelling scheme constitute a sequence of consecutive integers . Therefore, by Lemma 1, extends to an S--EAM total labelling of having magic constant .
Again from Theorem 1, we have a direct consequence as follows.

Theorem 15. For all positive integers, the pancyclic networkadmits an S--EAM total labelling.

2.4. S--EAM Total Labelling of Symmetric Lattice Networks

Definition 8. Consider a tripartite networkhaving vertex-edge connections as follows:In this section, we study the S--EAM total labelling of symmetric lattice networks , , and . These symmetric lattices contain copies of the tripartite network .

Definition 9. (i)For, .(ii)For.The lattice networkis a network with orderand sizedefined as follows:Figure 5 illustrates the general formation of the lattice network , .

Theorem 16. For all positive integers, the lattice networkis S--EAM total having magic constant.

Proof. (i)For : the vertex labelling extends to an S--EAM total labelling of , by Lemma 1.(ii)For .We are defining a labelling as follows:All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers . So, by Lemma 1, extends to an S--EAM total labelling of admitting magic constant .

Definition 10. (i)For, .(ii)For.The lattice networkhaving orderand sizeis defined as follows:In Figure 6, we have presented the general formation of the lattice network , .

Theorem 17. For all positive integers, the lattice networkis S--EAM total with magic constant.

Proof. (i)For : the vertex labelling extends to an S--EAM total labelling of , by Lemma 1.(ii)For .A labelling is defined as follows:All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers . So, by Lemma 1, extends to an S--EAM total labelling of having magic constant .

Definition 11. (i)For, .(ii)For.The lattice networkhaving orderand sizeis defined as follows:Figure 7 illustrates the general form of the lattice network , .

Theorem 18. For all positive integers, the lattice networkis S--EAM total with magic constant.

Proof. For , the labelling design is similar as in Theorem 17.

3. Illustration through Examples and Proposed Open Problems

3.1. Examples

The S--EAM total labelling of and S--EAM total labelling of are presented, respectively, in Figures 8(a) and 8(b). In Figure 8(a), the parameters are {, } and , while in Figure 8(b), the parameters are {, } and . These are perfect according to our depiction of the magic constants in the proofs of Theorems 2 and 3.

(a)

(b)

Similarly, Figures 9 and 10 illustrate Theorems 4 and 5 and Theorems 6 and 7, respectively, for the values of the parameters given in each. The values of here are perfectly similar as depicted in our main findings.

(a)

(b)

(a)

(b)

Figures 11(a) and 11(b) illustrate Theorems 11 and 12, respectively, for .

(a)

(b)

Figure 12 reveals an example of Theorem 15 corresponding to parameter .

Figures 13–15 refer to the illustration of S--EAM total labelling of lattice networks , , and (Theorems 16–18).

Due to facilitation of Lemma 1, edge labels are not needed to be provided in all of the above illustrative figures. As the edge sums constitute a sequence of + consecutive integers, assigning the remaining labels to the edges in ascending or descending order will generate S- or -EAM total labelling on that network, respectively, where (magic constant) and (minimum edge weight) attain some suitable values accordingly. More precisely, according to Lemma 1, this vertex labelling, consisting of consecutive integers, extends to an S--EAM total labelling of the networks.

3.2. Open Problems

The open problems relevant to the findings (Theorems 2–7) of Section 2 are proposed as follows:(i)For , obtain any S--EAM total labelling of the .(ii)For , obtain any S--EAM total labelling of .(iii)For , obtain any S--EAM total labelling of .(iv)For , determine S--EAM total labelling of , , , , , and for any other magic constants (i.e., for any other value of a) than computed here.(v)For , and positive integers, determine any S--EAM total labelling for the following networks:(1)(2)(3)

Open problems related to Theorems 11 and 12 are given as follows:(i)For , find any S--EAM total labelling of .(ii)For , find any S--EAM total labelling of .(iii)For , determine some S--EAM total labelling of and with a different magic constant than obtained here, i.e., for any other value of .

4. Synopsis and 3- Comparison of the Magic Constants and Minimum Edge Weights

This section consists of the synopsis (Section 4.1) and 3- graphical plots and comparison of the magic constants () and minimum edge weights () of our findings (Section 4.2).

4.1. Synopsis

Table 2 exhibits the computational results of our findings. The possible parameters for which we have determined S- and S--EAM total labellings are indicated through parameters column head.

4.2. Graphical Behavior of the Magic Constants and Minimum Edge Weights

Figure 16 shows the graphical comparison of the magic constants () and minimum edge weights (), in 3-, of the networks , , , , , , and , respectively. Moreover, Figure 16(a) shows that the most dominant layer is the one with green color. It interprets that among the magic constants of the networks discussed in this note, attains highest values with the increase of the values of the parameters.

(a)

(b)

Figure 17 illustrates the relative 3- comparison of the magic constants and minimum edge weights (corresponding to ) of the networks , , , , , , , and , for different values of the parameters.

(a)

(b)

(c)

(d)

(e)

(f)

5. Conclusion

In the present article,(i)We have designed S--EAM total labelling of the rooted product of cycle and complete bipartite network taking its disjoint union with paths and stars. The findings are related to the open problem on provided by Ngurah et al. in [48].(ii)We have provided S--EAM total labelling of rooted product of and pancyclic networks and .(iii)We have extended the result provided in [29] by Baig et al. through exhibiting S--EAM total labelling of pancyclic network involving chains of .(iv)We have exhibited S--EAM total labelling of symmetrically designed lattice networks , , and .(v)We have illustrated our findings through 3- graphical comparison.(vi)For further working in this field, several research problems have also been opened.(vii)The obtained schemes are now all set to serve as test ready labellings for programmers, networking professionals, and engineers to avail them where they find these appropriate.

Data Availability

The data used to support the findings of this study are included within this article. However, the reader may contact the corresponding author for more details on the data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

Z. Shao, “L (2,1)-labeling of the strong product of paths and cycles,” The Scientific Wolrd Journal, vol. 2014, Article ID 741932, 2014.
View at: Publisher Site | Google Scholar
T.-M. Wang and C.-C. Hsiao, “On anti-magic labeling for graph products,” Discrete Mathematics, vol. 308, no. 16, pp. 3624–3633, 2008.
View at: Publisher Site | Google Scholar
D. B. West, Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, NJ, USA, 2001.
J. Sedl, “Problem 27 in the theory of graphs and its applications,” in Proceedings of the Symposium, pp. 163–167, Smolenice, Slovakia, 1963.
View at: Google Scholar
N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academia Press, New York, NY, USA, 1990.
A. Kotzig and A. Rosa, “Magic valuations of finite graphs,” Canadian Mathematical Bulletin, vol. 13, no. 4, pp. 451–461, 1970.
View at: Publisher Site | Google Scholar
G. Ringel and A. S. Lldo, “Another tree conjecture,” Bull ICA, vol. 18, pp. 83–85, 1996.
View at: Google Scholar
H. Enomoto, A. Llado, T. Nakamigawa, and G. Ringel, “Super edge-magic graphs,” SUT Journal of Mathematics, vol. 34, pp. 105–109, 1998.
View at: Google Scholar
R. Simanjuntak, F. Bertault, and M. Miller, “Two new (a,d)- EAT graph labelings,” in Proceedings of the Eleventh Austrailian Workshop of Combinatorial Algorithm, pp. 179–189, Ballarat, Australia, 2000.
View at: Google Scholar
S. M. Lee and Q. X. Shan, All Trees with at Most 17 Vertices Are Super Edge- Magic, University Southern Illinois., Carbondale, Australia, 2002.
M. Javaid, A. A. Bhatti, and M. K. Aslam, “Super (a,d)-edge antimagic total labeling of a subclass of trees,” AKCE International Journal of Graphs and Combinatorics, vol. 14, no. 2, pp. 158–164, 2017.
View at: Publisher Site | Google Scholar
M. Javaid, “On super edge-antimagic total labeling of subdivided stars,” Discussiones Mathematicae Graph Theory, vol. 34, no. 4, pp. 691–705, 2014.
View at: Publisher Site | Google Scholar
M. Javaid and A. A. Bhatti, “On super edge- antimagic total labeling of subdivided stars,” Ars Combinatoria, vol. 105, pp. 503–512, 2012.
View at: Google Scholar
A. Raheem, A. Q. Baig, and M. Javaid, “On (a, d) -EAT labeling of subdivision of K1, r ,,” Journal of Information and Optimization Sciences, vol. 39, no. 3, pp. 643–650, 2018.
View at: Publisher Site | Google Scholar
A. N. M. Salman, A. A. G. Ngurah, and N. Izzati, “On super edge- magic total labeling of a subdivision of a star,” Utiltas Mathematica, vol. 81, pp. 275–284, 2010.
View at: Google Scholar
M. Javaid and A. A. Bhatti, “On super edge- antimagic total labeling of generalized extended -trees,” AKCE International Journal of Graphs and Combinatorics, vol. 11, pp. 115–126, 2014.
View at: Google Scholar
M. Javaid, A. A. Bhatti, and M. Hussain, “On edge- antimagic total labeling of extended -trees,” Utilitas Math, vol. 87, pp. 293–303, 2012.
View at: Google Scholar
M. Javaid, M. Hussain, K. Ali, and K. H. Dar, “Super edge-magic total labeling on -trees,” Utilitas Math, vol. 86, pp. 183–191, 2011.
View at: Google Scholar
M. Hussain and E. T. Boskoro, “Slamin, on super edge magic total labeling of banana trees,” Utiltas Mathematica, vol. 79, pp. 243–251, 2009.
View at: Google Scholar
K. A. Sugeng, M. Miller, and M. Baa, “edge- antimagic total labelings of caterpillars,” Lecture Notes in Computer Science, vol. 3330, Article ID 169aAS180, 2005.
View at: Google Scholar
M. Javaid, A. A. Bhatti, and M. Hussain, “On super edge- antimagic total labeling of subdivided caterpillar,” Utilitas Math, vol. 98, pp. 227–241, 2015.
View at: Google Scholar
R. M. Figueroa-Centeno, R. Ichishima, and F. A. Muntaner-Batle, “On super edge- magic graph,” Ars Combinatoria, vol. 64, pp. 81–95, 2002.
View at: Google Scholar
A. A. Bhatti, A. Nisar, and M. Kanwal, “Radio number of wheel like graphs,” International Journal on Applications of Graph Theory In Wireless Ad Hoc Networks And Sensor Networks, vol. 3, no. 4, pp. 39–57, 2011.
View at: Publisher Site | Google Scholar
J. A. Gallian, “A dynamic survey of graph labeling,” The Electronic Journal of Combinatorics, 2019.
View at: Google Scholar
R. Khennoufa and O. Togni, “The radio antipodal and radio numbers of the hypercube,” Ars Combinatoria, vol. 102, pp. 447–461, 2011.
View at: Google Scholar
R. Figueroa-Centeno, R. Ichishima, and F. Muntaner-Batle, “On edge-magic labelings of certain disjoint unions of graphs,” Australas. J. Combin., vol. 32, pp. 225–242, 2005.
View at: Google Scholar
R. M. Figueroa-Centeno, R. Ichishima, F. A. Muntaner-Batle, and A. Oshima, “A magical approach to some labeling conjectures,” Discussiones Mathematicae Graph Theory, vol. 31, no. 1, pp. 79–113, 2011.
View at: Publisher Site | Google Scholar
R. M. Figueroa-Centeno, R. Ichishima, and F. A. Muntaner-Batle, “The place of super edge-magic labelings among other classes of labelings,” Discrete Mathematics, vol. 231, no. 1-3, pp. 153–168, 2001.
View at: Publisher Site | Google Scholar
A. Q. Baig, H. U. Afzal, M. Imran, and I. Javaid, “Super edge- magic labeling of volvox and pancyclic graphs,” Util. Math., vol. 93, pp. 49–56, 2014.
View at: Google Scholar
J.-B. Liu, M. K. Aslam, M. Javaid, and A. Raheem, “Computing edge-weight bounds of antimagic labeling on a class of trees,” IEEE Access, vol. 7, pp. 93375–93386, 2019.
View at: Publisher Site | Google Scholar
A. Ahmad, M. F. Nadeem, and A. Gupta, “On super edge- magic deficiency of certain Toeplitz graphs,” Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 3, pp. 1–7, 2018.
View at: Google Scholar
A. Ahmad, K. Ali, M. Bača, P. Kovář, and A. Semaničová-Feňovčíková, “Vertex-antimagic labelings of regular graphs,” Acta Mathematica Sinica, English Series, vol. 28, no. 9, pp. 1865–1874, 2012.
View at: Publisher Site | Google Scholar
M. Bača and M. K. Siddiqui, “Construction of tree from smaller graceful trees,” Utilitas Mathematica, vol. 99, no. 1, pp. 175–186, 2016.
View at: Google Scholar
A. Ahmad and F. A. Muntaner-Batle, “On super edge magic deficiency of unicyclic graphs,” Utilitas Math, vol. 98, pp. 379–386, 2015.
View at: Google Scholar
A. Ahmad, A. Q. Baig, and M. Imran, “On super edge-magicness of graphs,” Utilitas Math, vol. 89, pp. 373–380, 2012.
View at: Google Scholar
P. R. L. Pushpam and A. Saibulla, “Super edge- antimagic total labeling of some classes of graphs,” SUT Journal of Mathematics, vol. 48, no. 1, pp. 1–12, 2012.
View at: Google Scholar
N. L. Prasanna, K. Sravanthi, and N. Sudhakar, “Applications of graph labeling in major areas of computer science,” International Journal of Research in Computer and Communication Technology, vol. 3, pp. 819–823, 2014.
View at: Google Scholar
N. L. Prasanna, “Applications of graph labeling in communication networks,” Oriental Journal of Computer Science and Technology, vol. 7, pp. 893–897, 2014.
View at: Google Scholar
M. Andreas, Wireless Communications, Wiley-IEEE, New York, NY, USA, 2005.
R. Wakefield, Radio Broadcasting, W2UC, Union College, Schenectady, NY, USA, 1959.
Z. Zhou, S. Das, and H. Gupta, “Connected K-coverage problem in sensor networks,” in Proceedings. 13th International Conference on Computer Communications and Networks, pp. 373–378, IEEE Cat. No.04EX969), Chicago, IL, USA, 2004.
View at: Publisher Site | Google Scholar
M. S. Vinutha and P. Arathi, “Applications of graph coloring and labeling in computer science,” International Journal on Future Revolution in Computer Science and Communication Engineering, vol. 3, pp. 14-15, 2017.
View at: Google Scholar
I. Katzela and M. Naghshineh, “Channel assignment schemes for cellular mobile telecommunication systems: a comprehensive survey,” IEEE Personal Communications, vol. 3, no. 3, pp. 10–31, 1996.
View at: Publisher Site | Google Scholar
A. Krishnaa, “On antimagic labellings of some cycle related graphs,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 15, pp. 225–235, 2012.
View at: Publisher Site | Google Scholar
A. Krishnaa, “Formulas and algorithms of antimagic labelings of Some helm related graphs,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 19, no. 2, pp. 435–445, 2016.
View at: Publisher Site | Google Scholar
A. Krishnaa, “Some applications of labelled graphs,” International Journal of Mathematics Trends and Technology, vol. 37, no. 3, pp. 19–23, 2016.
View at: Publisher Site | Google Scholar
J. P. Desai, “A graph theoretic approach for modeling mobile robot team formations,” Journal of Robotic Systems, vol. 19, no. 11, pp. 511–525, 2002.
View at: Publisher Site | Google Scholar
A. A. G. Ngurah, E. T. Baskoro, R. Simanjuntak, and S. Uttunggadewa, “On the super edge-magic strength and deficiency of graphs,” Kyoto CGGT, vol. 4535, pp. 144–154, 2008.
View at: Google Scholar

Copyright

Copyright © 2021 Hafiz Usman Afzal 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.

PDF Download Citation

Download other formats

Order printed copies

Views

272

Downloads

616

Citations