Research Computing Edge Weights of Symmetric Classes of Networks

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 ( a, 0 ) edge-antimagic labelling on networks was identiﬁed in the late nineties. The present article addresses super ( a, 0 ) edge antimagicness of union of the networks’ star S n , the path P n , and copies of paths and the rooted product of cycle C n with K 2 ,m . We also provide super ( a, 0 ) edge-antimagic labelling of the rooted product of cycle C n and planar pancyclic networks. Further, we design a super ( a, 0 ) edge-antimagic labelling on a pancyclic network containing chains of C 6 and three diﬀerent symmetrically designed lattices. Moreover, our ﬁndings have also been recapitulated in the shape of 3- D plots and tables.


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.

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). e present article addresses super (a, 0) edge-antimagic labelling of the rooted product of K 2,m and C n taking its disjoint union with the star, path, and copies of paths. We shall also design super (a, 0) edgeantimagic labelling on rooted product of specifically designed planar pancyclic networks with cycle C n . Moreover, we shall design super (a, 0) edge-antimagic labelling on planar pancyclic networks containing chains of C 6 and three different symmetric lattice networks (notated as L 1 n , L 2 n , and L 3 n ). Except lattice networks, interestingly, all networks discussed in this note are planar. e 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. e 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. Definition 4. Let G 1 and G 2 be two simple networks. e network obtained by taking |V(G 1 )| copies of G 2 and then for each point (vertex) v j in V (called the root vertex) (G 1 ), v j is being replaced with the j th copy of G 2 , termed as the rooted product of the networks G 1 and G 2 . It is notated as G 1°G2 .
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.
e idea of magic labelling on networks was identified by Sadlácek in 1963 [4]. e 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 (a, 0)-EAM total labelling on networks (studied by Ringel and Llado [7] in 1996). e notion of S-(a, 0)-EAM total labelling of networks was defined by Enomoto et al. [8] with the terminology super edge-magic labelling. Simanjantuk et al. highlighted (a, d)-EAM total labelling of networks in [9] in year 2000. e literature of (a, 0)-EAM total labelling of networks includes the following interesting and useful conjectures.

Conjecture 2. All trees admit S-(a, 0)-EAM total labelling
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], w-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][24][25]. However, this conjecture is still open for working. Enomoto et al. proved that if a simple (p, q)-network G is S-(a, 0)-EAM total, then 2p − 3 is at least q [8]. ey further derived that the network K m,n is S-(a, 0)-EAM total ⇔ m or n is 1. Figueroa-Centeno et al. derived that the union of networks K 1,m ∪ K 1,n is S-(a, 0)-EAM total if either m � η 1 (n + 1) or n � η 2 (n + 1) [26]. e network C n is also proved to be S-(a, 0)-EAM total only if n ≡ 1(mod 2) in [8]. In [27], C 3 ∪ C n has been proven to be S-(a, 0)-EAM total only when 6 ≤ n ≡ 0(mod 2). e generalized prism D m,n is proven to be S-(a, 0)-EAM total for all odd values of m in [28]. Baig et al. classified a class of planar pancyclic networks in [29] and exhibited its S-(a, 0)-EAM total labelling for all possible values of the parameters involved. An immensely advantageous lemma on S-(a, 0)-EAM total networks is as follows. Liu at al. studied the bounds of the minimum and maximum edge weights for super (a, d)-EAM total labelling on a generalized class of subdivided caterpillars in [30] for various values of d. In [31], Ahmad et al. studied the super (a, 0)-EAM total labelling of certain Toeplitz graphs combined with isolated vertices nK 1 , for various values of n (also known as super edge-magic deficiency of networks). e properties and existence of super (a, d) vertex-antimagic labelling of regular graphs have been discussed in [32]. In [33], Ahmad et al. constructed the α-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-(a, 0)-EAM total labelling on the graphs G ∪ nK 1 has been studied, where G represents the unicyclic graph, whereas S-(a, 0)-EAM total labelling of networks like zig-zag triangle and disjoint union of combs and stars has been studied in [35].
In Lemma 1, the sum δ(x) + δ(y) is called as edge sum for each edge xy ∈ E(G). is 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-(a, 0)-EAM total, if the edge sums are positive consecutive integers. e following result is also very pertinent as far as S-(a, 0)-EAM total networks are concerned. [36].

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. e 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]. e study of magic labelling has been appearing to be more useful gradually in the data mining. e 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.

Networking.
e 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. e importance and large-scale usage of wired networking cannot be denied in the present era as well.
e 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]). e modern era is of network communication whose part and parcel is radio transmission. e interference, making the job of channel assignment more complicated, is one of the major concerns in radio transmission. e transmission of concurrent networks that are constraint-free, admitting same instance surfacing, is the central reason of this unwanted interruption [39,40]. e magic labelling assists in the allotment of constant weights to the networks that are concurrent. Such interferences are eliminated by using this procedure. e 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 (a, 0)-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]).

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. e defiance task here for the base cell is to facilitate with the ability to re-use utmost channels, avoiding any violation of the constraints. is 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. e label of the path specifies uniquely the two users which it interlinks conversely (see [43]).

Urban Planning.
Consider the wheel W 6 , the helm H 6 , and prism D 5 in Figure 1 as a specific example. e edges of the networks W 6 and H 6 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 D 5 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. e magic constant, in the scenario of design like D 5 , gets disordered promptly in case of violation in the pathway.
is 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. at is, as a model for surveillance of the extensively secure areas, these labellings perform their distinct role [46].

Robotics.
e 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. e 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.

Main Results
is section contains our main findings. It is divided into four subsections further. In Section 2.1, the S-(a, 0)-EAM total labelling on the union of K 2,m ∘ C n with copies of paths, the star, and the path shall be designed, whereas in Section 2.2, we derive an S-(a, 0)-EAM total labelling on the rooted product of network C n and pancyclic networks H 1 and H 2 (planar also). Further, S-(a, 0)-EAM total labelling of planar pancyclic network Γ n and symmetric lattice networks L 1 n , L 2 n , and L 3 n shall be exhibited in Sections 2.3 and 2.4, respectively.

S-(a, 0)-EAM Total
Labelling of the Disjoint Union of C n°K2,m and P n , S n , and mP 2 . Our main motivation to explore the findings in this section is the following open problem of Ngurah et al. [48].
Open Problem. For n ≥ 2 and m ≥ 3, is there any S-(a, 0)-EAM total labelling of nK 2,m ?
In fact, the rooted product C n°K2,m contains n copies of the complete bipartite network K 2,m . A cycle C n is a 2regular network of order n, whereas the complete bipartite network K 2,m is class-wise regular in which one partitioned class of vertices is 2-regular and other is m-regular, where |V(K 2,m )| � m + 2.

S-(a, 0)-EAM Total Labelling of Rooted Product of Pancyclic Networks with C n .
e present section deals with S-(a, 0)-EAM total labelling of the rooted product of two specific planar non-isomorphic pancyclic networks and the cycle C n . A specific pancyclic network H 1 is defined as follows.
Definition 5. H 1 is a pancyclic network having the following construction.
Consider the network C n ∘ H 1 with |V(C n ∘ H 1 )| � 8n and |E(C n ∘ H 1 )| � 14n connected as per the following scheme:
Consider C n ∘ H 2 with |V(C n ∘ H 2 )| � 8n and |E(C n ∘ H 2 )| � 14n with the following connection: e labelling scheme for n � 1 and n ≥ 3 is the same as ψ 1 designed in eorem 11.
A direct derivation from eorem 1 is given as follows.
Theorem 13. For n ≡ 1(mod2), C n ∘ H 1 and C n ∘ H 2 are S-((17n + 5/2), 2)-EAM total. [29]. In [29], Baig et al. provided a result regarding S-(a, 0)-EAM total labelling of a pancyclic class of networks involving chains of cycle C 4 . Here, we shall introduce a pancyclic family of networks involving chains of cycle C 6 , while our point of convergence is the S-(a, 0)-EAM total labelling of this class. us, we further extend the results of Baig et al. [29].

S-(a, 0)-EAM Total Labelling of a Pancyclic Class of Networks: Extension of a Result Appearing in
Definition 7. e pancyclic network Γ n is a network with order |V(Γ n )| � 6n and |E(Γ n )| � 12n − 3, with structure as follows: Figure 3 reveals general formation of Γ n . In Figure 4, we have shown the network Γ 2 and its contained cycles of orders 3, 4, . . . , 12. In the upcoming result, we show that the pancyclic network Γ n is S-(a, 0)-EAM total.

Theorem 14.
For all positive integers n, the pancyclic network Γ n is S-(a, 0)-EAM total having magic constant 18n.
Define here a labelling g: All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers 3, 4, . . . , 12n − 1. erefore, by Lemma 1, g extends to an S-(a, 0)-EAM total labelling of Γ n having magic constant a � 18n.
Again from eorem 1, we have a direct consequence as follows.
In this section, we study the S-(a, 0)-EAM total labelling of symmetric lattice networks L 1 n , L 2 n , and L 3 n . ese symmetric lattices contain n copies of the tripartite network T.

Definition 9
(i) For n � 1, e lattice network L 1 n is a network with order 6n and size 12n − 3 defined as follows: Figure 5 illustrates the general formation of the lattice network L 1 n , ∀ n ∈ N.
Theorem 16. For all positive integers n, the lattice network L 1 n is S-(a, 0)-EAM total having magic constant 18n.

Definition 10
(i) For n � 1, L 2 1 � T. (ii) For n ≥ 2.  e lattice network L 2 n having order 10n − 4 and size 20n − 11 is defined as follows: In Figure 6, we have presented the general formation of the lattice network L 2 n , ∀ n ∈ N.
Theorem 17. For all positive integers n, the lattice network L 2 n is S-(a, 0)-EAM total with magic constant 30n − 12.

Definition 11
(i) For n � 1, L 3 1 � T. (ii) For n ≥ 2. e lattice network L 3 n having order 10n − 4 and size 20n − 11 is defined as follows: Figure 7 illustrates the general form of the lattice network L 3 n , ∀ n ∈ N.
Theorem 18. For all positive integers n, the lattice network L 3 n is S-(a, 0)-EAM total with magic constant 30n − 12.
Proof. For L 3 n , the labelling design is similar as in eorem 17.

Illustration through Examples and Proposed
Open Problems Examples. e S-(132, 0)-EAM total labelling of (C 5 ∘ K 2,6 ) ∪ K 1,5 ∪ 2K 1 and S-(163, 0)-EAM total labelling of (C 7 ∘ K 2,5 ) ∪ K 1,7 ∪ 3K 1 are presented, respectively, in  Figure 12 reveals an example of eorem 15 corresponding to parameter n � 3. Figures 13-15 refer to the illustration of S-(a, 0)-EAM total labelling of lattice networks L 1 5 , L 2 4 , and L 3 4 ( eorems [16][17][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 +ve consecutive integers, assigning the remaining labels q, q − 1, . . . , p + 2, p + 1 to the edges in ascending or descending order will generate S-(a ′ , 2) or (a, 0)-EAM total labelling on that network, respectively, where a (magic constant) and a ′ (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-(a, 0)-EAM total labelling of the networks.

Conclusion
In the present article, (i) We have designed S-(a, 0)-EAM total labelling of the rooted product of cycle C n and complete bipartite network K 2,m taking its disjoint union with paths and stars. e findings are related to the open problem on nK 2,m provided by Ngurah et al. in [48]. (ii) We have provided S-(a, 0)-EAM total labelling of rooted product of C n and pancyclic networks H 1 and H 2 . (iii) We have extended the result provided in [29] by Baig et al. through exhibiting S-(a, 0)-EAM total labelling of pancyclic network Γ n involving chains of C 6 . (iv) We have exhibited S-(a, 0)-EAM total labelling of symmetrically designed lattice networks L 1 n , L 2 n , and L 3 n . (v) We have illustrated our findings through 3-D graphical comparison. (vi) For further working in this field, several research problems have also been opened. (vii) e 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
e 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
e authors declare that they have no conflicts of interest.