Computing Edge Weights of Magic Labeling on Rooted Products of Graphs

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 andmagic graphs serve asmodels for surveillance or security systems in urban planning. In 1998, Enomoto et al. introduced the notion of super (a, 0) edge-antimagic labeling of graphs. In this article, we shall compute super (a, 0) edge-antimagic labeling of the rooted product of Pn and the complete bipartite graph (K2,m) combined with the union of path, copies of paths, and the star. We shall also compute a super (a, 0) edgeantimagic labeling of rooted product of Pn with a special type of pancyclic graphs. (e labeling provided here will also serve as super (a′, 2) 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 3D plots.


Introduction
e 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][2][3][4][5]. In this article, we will provide super (a, 0) edgeantimagic labeling of the rooted product of P n and the complete bipartite graph (K 2,m ) taking its disjoint union with path, copies of paths, and star graph. Further, we will provide super (a, 0) edge-antimagic labeling of the rooted product of path P n with specifically designed pancyclic graphs. We shall, in particular, target the planar graphs that are obtained as a result of our rooted products. ese planar graphs minimize the possibility of overlapping of various entities in practical purposes, which is a major cause of inefficiency in organizations. e super (a, 0) 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.

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.

Networking. A network consists of nodes (vertices) and
links between these nodes (edges). More technically, a network, say N, is an ordered 2-tuple consisting of two sets, set of nodes V(N) and links between nodes called set of edges E(N) such that (E(N) ⊂ V(N) × V(N)).
us, 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. ese wireless radio networks face a major challenge in the form of interference which makes the task of channel assignment harder. e 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. is 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. e (a, 0) edgeantimagic 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. is labeling is designed with a constant edge weight, which helps routing to automatically detect the next node within that network (see [10]).

Telecommunication.
e 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.
e base cell has the ability to communicate with mobile stations using its radio transceiver. e 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]).

Civil Engineering and Urban
Planning. As a particular example, consider the wheel W 6 , helm H 6 , and prisms D 5 and D 6 in Figure 1. e edges of the graphs W 6 , H 6 , and D 6 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. at is, the vertex antimagic labeling of these graphs is being provided here. Meanwhile, on D 5 , 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 D 5 , the magic constant will get disturbed abruptly due to violation of passage. is 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. is is a major use of magic labeling and antimagic labeling in urban planning.
us, these labelings play their part as model for surveillance or security for various types of buildings or areas [15].
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. e 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 3D plots of our results. Conclusion is given in the last section.

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 G � (V(G), E(G)) be a simple, connected, and nonempty graph with vertex set V(G) and edge set E(G) such that |V(G)| � p and |E(G)| � q, respectively. We call G in this case a (p, q)graph. roughout this article, we will use (p, q)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 2 Mathematical Problems in Engineering 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.
for each xy ∈ E(G), form a sequence of consecutive positive integers with minimum edge weight (a > 0) and common difference d. If such a labeling exists, then G is said to be an (a, d) edge-antimagic graph.
In the above definitions, if we have d � 0, then the minimum edge weight a becomes constant for all edges xy ∈ E(G) known as magic constant or magic sum of the graph G.  e rooted product of two graphs G 1 and G 2 is obtained by taking |V(G 1 )| copies of G 2 and then, for every vertex v i of G 1 , identifying v i with the root node of the i th copy of G 2 . It is denoted by (G 1 ∘ G 2 ).
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. e concept of (a, 0) 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, (a, 0) edge-antimagic labeling. Motivated by this concept, Enomoto et al., in 1998, defined the notion of super (a, 0) edge-antimagic labeling of graphs in [22]. ey studied this concept with the term super edge-magic labeling of graphs. In the year 2000, Simanjantuk et al. introduced the idea of (a, d) edge-antimagic labeling of graphs in [23]. e historical background of the (a, 0) edge-antimagic labeling of graphs includes the following important and interesting conjectures on trees.
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][26][27][28][29], W-trees [30][31][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][38][39]. However, this conjecture is still open for working. In [22], it is proven that if a nontrivial (p, q)graph G is super (a, 0) edge-antimagic, then (q ≤ 2p − 3). In the same article, the authors proved that a complete bipartite graph K m,n is super (a, 0) edge-antimagic if and only if m � 1 or n � 1. In [40], it is proven that K 1,m ∪ K 1,n is super (a, 0) edge-antimagic if either m is a multiple of n + 1 or n is a multiple of m + 1. Enomoto et al. [22] proved that C n is super (a, 0) edge-antimagic if and only if n is odd. In [41], it has been proven that C 3 ∪ C n is super (a, 0) edge-antimagic if and only if n ≥ 6 and n is even (also see [42]). In [43], Figueroa-Centeno et al. showed that the generalized prism C n × P m is super (a, 0) edge-antimagic for every odd integer n. In [3], Baig et al. presented the super (a, 0) edge-antimagic labeling of a class of pancyclic graphs. e following lemma regarding super (a, 0) edge-antimagic graphs is very useful.
Mathematical Problems in Engineering antimagic vertex labeling of the graph G [23]. at is, f constitutes a super (a, 1) edge-antimagic vertex labeling of G. It means Lemma 1 states that whenever the set of edge sums forms a super (a, 1) edge-antimagic vertex labeling of G, it extends to a super (a, 0) edge-antimagic labeling of G.
We will frequently use the above lemma in the proofs of our results. Another very useful relevant result is as follows.

Main Results
is section consists of two further sections, in which we will present our main results. In Section 3.1, we will study the super (a, 0) edge-antimagic labeling of the disjoint union of the rooted product of P n and the complete bipartite graph K 2,m with path, copies of paths, and the star. Meanwhile, in Section 3.2, we will provide the super (a, 0) edge-antimagic labeling of rooted product of P n with certain pancyclic graphs H 1 and H 2 . It is pertinent to mention here that all our graphs obtained as the result of the rooted products are planar. e following open problem proposed by Ngurah et al. in [46] is our main motivation to study super (a, 0) edge-antimagic labeling of the graph containing copies of complete bipartite graph K 2,m .

Open Problem.
For n ≥ 2 and m ≥ 3, can you determine any super (a, 0) edge-antimagic labeling of (nK 2,m )? e rooted product (P n ∘ K 2,m ), in fact, contains n copies of the complete bipartite graph K 2,m . We swiftly move to our theorems now.
Proof. Consider the graph (G 6 � (P n ∘ K 2,m ) ∪ P n+1 ) for both odd m and n ≥ 3 with vertex and edge sets as follows.
Theorem 10. For all m ≥ 2 and odd n ≥ 3, the graph ((P n°K2,m ) ∪ P n+1 ) admits a super (((2mn + 9n + 11)/2), 2) edge-antimagic labeling. (a, 0) Edge-Antimagic Labeling of Rooted Product of P n 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. is kind of network corresponds to a pancyclic graph (see Definition 3).

Super
We define here a specific pancyclic graph H 1 as follows.
Definition 5. Consider a pancyclic graph H 1 with the construction: In the next result, we will show that the rooted product (P n ∘ H 1 ) is super (a, 0) edge-antimagic for odd values of n.
Proof. Consider the rooted product (P n ∘ H 2 ) with (|V(P n ∘ H 2 )| � 8n) and (|E(P n ∘ H 2 )| � 14n − 1) with vertex and edge sets as follows: And the labeling scheme is the same as ψ 1 designed in eorem 11. e following result is a direct consequence of eorem 1.

Examples.
As examples of eorems 2 and 3, the super (a, 0) edge-antimagic labeling of ((P 5 ∘ K 2,6 ) ∪ K 1,5 ∪ 2K 1 ) and super (a, 0) edge-antimagic labeling of ((P 5 ∘ K 2,7 ) ∪ K 1,5 ∪ 2K 1 ) 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 n � 5 and m � 6) and the magic constant of the labeling in Figure 4 is 147 (corresponding to the parameters n � 5 and m � 7). ese are as per the magic constants depicted in the proofs of eorems 2 and 3.
Similarly, Figures 5-8 illustrate eorems 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 (114, 0) edge-antimagic labeling of the rooted products appearing in eorems 11 and 12 with parameter n � 5.
(2) For even n ≥ 2, can you determine any super (a, 0) edge-antimagic labeling of ((P n ∘ K 2,m )) ∪ nP 2 )? (3) For even n ≥ 2, can you determine any super (a, 0) edge-antimagic labeling of ((P n ∘ K 2,m ) ∪ P n+1 )? (4) For odd n ≥ 3, can you determine super (a, 0) edgeantimagic labeling of the graphs G 1 , G 2 , G 3 , G 4 , G 5 , and G 6 for a magic constant (i.e., for any other value of a ) other than those obtained in this article? (5) For l, m, n ∈ N, find any super (a, 0) edge-antimagic labeling of the following graphs: e open problems related to the results ( eorems 11 and 12) presented in Section 3.2 are as follows: (1) For even n ≥ 2, can you determine any super (a, 0) edge-antimagic labeling of the rooted product (P n ∘ H 1 )? (2) For even n ≥ 2, can you determine any super (a, 0) edge-antimagic labeling of the rooted product (P n ∘ H 2 )? (3) For odd n ≥ 3, can you determine any super (a, 0) edge-antimagic labeling of the rooted products (P n ∘ H 1 ) and (P n ∘ H 2 ) with a magic constant that is different from those obtained here, that is, for any other value of a

Synopsis.
e computational results of our findings have been summarized in Table 1.
e "Parameters" column represents those parameters for which we have been able to find super (a, 0) and (a ′ , 2) edge-antimagic labeling of the corresponding graph.

3D Graphical Comparison.
e following are the 3D plot representations of the magic constants obtained in our  Figure 9: A super (a, 0) edge-antimagic labeling of (P 5 ∘ H 1 ) with magic constant a � 114.

Conclusion
In this note, (1) we have obtained a super (a, 0) edge-antimagic labeling of the rooted product of P n with the complete bipartite graph (K 2,m ) taking its union with path, copies of paths, and star for possible values of n. e obtained results partially point towards the problem proposed in [46] regarding nK 2,m ; (2) we have presented a super (a, 0) edge-antimagic labeling of the rooted product of P n with special pancyclic graphs H 1 and H 2 ; (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) e obtained schemes are now all set to serve as testready labelings for engineers, programmers, and networking professionals to utilize them where they find them suitable.
Data Availability e whole data are included within this article. However, more details on the data can be obtained from the corresponding author upon request.   Figure 11: e 3D plots of the magic constants of M 1 � (P n ∘ K 2,m ) ∪ K 1,n ∪ (n − 1/2)K 1 , M 2 � (P n ∘ K 2,m ) ∪ nP 2 , (M 3 � (P n ∘ K 2,m ) ∪ P n+1 ), and (P n ∘ H 1 ) and (P n ∘ H 2 ).