# Computing Edge Weights of Symmetric Classes of Networks

Authors: Hafiz Usman Afzal; Muhammad Javaid; Abdulaziz Mohammed Alanazi; Maryam Gharamah Alshehri
Journal: Mathematical Problems in Engineering
 (2021)

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## 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 supera,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 Sn, the path Pn, and copies of paths and the rooted product of cycle Cn with K2,m. We also provide super a,0 edge-antimagic labelling of the rooted product of cycle Cn and planar pancyclic networks. Further, we design a super a,0 edge-antimagic labelling on a pancyclic network containing chains of C6 and three different symmetrically designed lattices. Moreover, our findings have also been recapitulated in the shape of 3-D plots and tables.


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## Body

## 1. Introduction

In this section, we shall define our problem and explain the objective of this study in Section1.1, 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 a,0 edge-antimagic labelling of the rooted product of K2,m and Cn taking its disjoint union with the star, path, and copies of paths. We shall also design super a,0 edge-antimagic labelling on rooted product of specifically designed planar pancyclic networks with cycle Cn. Moreover, we shall design super a,0 edge-antimagic labelling on planar pancyclic networks containing chains of C6 and three different symmetric lattice networks (notated as Ln1, Ln2, and Ln3). 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 setVG and connection between these vertices termed as edge set EG, is called a network G, where EG is contained in VG×VG. A network G can either be connected or consists of connected components. We will consider nonempty and simple networks throughout, having VG as its vertex set and EG as its edge set with order VG=p and size EG=q. The network G in this case is termed as a p,q-network. Reference [3] is referred to for further discernment into the network terminologies.A labelling is a function that maps +ive 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, VG or EG 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 ap,q-networkG=VG,EG, the bijective functionδformVG∪EGonto1,2,…,p+qis termed asa,dedge-antimagic total labelling with the constraint that the edge weightsδx+δxy+δy, for eachxy∈EG, constitute a sequence of consecutive positive integers, whereais the minimum edge weight and common difference isd.Gis referred to be ana,dedge-antimagic total network, if such a labelling exists.Definition 2.
A supera,dedge-antimagic total labelling is in fact ana,dedge-antimagic total labelling in which minimum labels1,2,…,pare assigned to vertices of thep,q-networkG.Gis termed to be a supera,dedge-antimagic total network in this case.
In Definitions1 and 2, the minimum edge weight a becomes constant c at d=0, for all edges xy∈EG, which is referred to as magic sum or magic constant for the network G.Definition 3.
A networkGis termed as a pancyclic network if it contains cycle of every order from 3 toVG.Definition 4.
LetG1andG2be two simple networks. The network obtained by taking VG1copies ofG2andthen for each point (vertex)vj in V (called the root vertex) (G1), vj is being replaced with the jthcopy ofG2, termed as the rooted product of the networks G1andG2. It is notated asG1°G2.
Further in the article, the abbreviations are being used as given in. Table1.
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 a,0-EAM total labelling on networks (studied by Ringel and Llado [7] in 1996). The 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.
The literature ofa,0-EAM total labelling of networks includes the following interesting and useful conjectures.Table 1
Notations and their abbreviations used in the article.
TerminologyAbbreviationa,d edge-antimagica,d-EAMSupera,d edge-antimagicS-a,d-EAMConjecture 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 Conjecture2, 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 [23–25]. 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]. They further derived that the network Km,n is S-a,0-EAM total ⇔m or n is 1. Figueroa-Centeno et al. derived that the union of networks K1,m∪K1,n is S-a,0-EAM total if either m=η1n+1 or n=η2n+1 [26]. The network Cn is also proved to be S-a,0-EAM total only if n≡1mod2 in [8]. In [27], C3∪Cn has been proven to be S-a,0-EAM total only when 6≤n≡0mod2. The generalized prism Dm,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 nK1, for various values of n (also known as super edge-magic deficiency of networks). The 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∪nK1 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].Lemma 1 (see [28]).
A℘,ℑ-networkGis S-a,0-EAM total if and only if there is a bijectionδ:VG⟶1,2,…,℘such that the setS=δx+δy|xy∈EGconsists ofℑconsecutive integers. In such a case,δextends to an S-a,0-EAM total labelling ofGwith magic constantℜ=℘+ℑ+s, wheres=minSandS=ℜ−℘+1,ℜ−℘+2,…,ℜ−℘+ℑ.In Lemma1, the sum δx+δy is called as edge sum for each edge xy∈EG. 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-a,0-EAM total, if the edge sums are positive consecutive integers. The following result is also very pertinent as far as S-a,0-EAM total networks are concerned.Theorem 1.
A simple network G admits an S-(a, 0)-EAM total labelling⇔G admits an S-(a-EG+ 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 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]).


#### 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 wheelW6, the helm H6, and prism D5 in Figure 1 as a specific example. The edges of the networks W6 and H6 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 D5 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 D5, 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].Figure 1
Vertex-antimagic total labelling of the wheelW6 and helm H6 and edge-magic labelling total of the prism D5 [44,45].
(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.


## 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 a,0 edge-antimagic labelling of the rooted product of K2,m and Cn taking its disjoint union with the star, path, and copies of paths. We shall also design super a,0 edge-antimagic labelling on rooted product of specifically designed planar pancyclic networks with cycle Cn. Moreover, we shall design super a,0 edge-antimagic labelling on planar pancyclic networks containing chains of C6 and three different symmetric lattice networks (notated as Ln1, Ln2, and Ln3). 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 setVG and connection between these vertices termed as edge set EG, is called a network G, where EG is contained in VG×VG. A network G can either be connected or consists of connected components. We will consider nonempty and simple networks throughout, having VG as its vertex set and EG as its edge set with order VG=p and size EG=q. The network G in this case is termed as a p,q-network. Reference [3] is referred to for further discernment into the network terminologies.A labelling is a function that maps +ive 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, VG or EG 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 ap,q-networkG=VG,EG, the bijective functionδformVG∪EGonto1,2,…,p+qis termed asa,dedge-antimagic total labelling with the constraint that the edge weightsδx+δxy+δy, for eachxy∈EG, constitute a sequence of consecutive positive integers, whereais the minimum edge weight and common difference isd.Gis referred to be ana,dedge-antimagic total network, if such a labelling exists.Definition 2.
A supera,dedge-antimagic total labelling is in fact ana,dedge-antimagic total labelling in which minimum labels1,2,…,pare assigned to vertices of thep,q-networkG.Gis termed to be a supera,dedge-antimagic total network in this case.
In Definitions1 and 2, the minimum edge weight a becomes constant c at d=0, for all edges xy∈EG, which is referred to as magic sum or magic constant for the network G.Definition 3.
A networkGis termed as a pancyclic network if it contains cycle of every order from 3 toVG.Definition 4.
LetG1andG2be two simple networks. The network obtained by taking VG1copies ofG2andthen for each point (vertex)vj in V (called the root vertex) (G1), vj is being replaced with the jthcopy ofG2, termed as the rooted product of the networks G1andG2. It is notated asG1°G2.
Further in the article, the abbreviations are being used as given in. Table1.
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 a,0-EAM total labelling on networks (studied by Ringel and Llado [7] in 1996). The 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.
The literature ofa,0-EAM total labelling of networks includes the following interesting and useful conjectures.Table 1
Notations and their abbreviations used in the article.
TerminologyAbbreviationa,d edge-antimagica,d-EAMSupera,d edge-antimagicS-a,d-EAMConjecture 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 Conjecture2, 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 [23–25]. 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]. They further derived that the network Km,n is S-a,0-EAM total ⇔m or n is 1. Figueroa-Centeno et al. derived that the union of networks K1,m∪K1,n is S-a,0-EAM total if either m=η1n+1 or n=η2n+1 [26]. The network Cn is also proved to be S-a,0-EAM total only if n≡1mod2 in [8]. In [27], C3∪Cn has been proven to be S-a,0-EAM total only when 6≤n≡0mod2. The generalized prism Dm,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 nK1, for various values of n (also known as super edge-magic deficiency of networks). The 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∪nK1 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].Lemma 1 (see [28]).
A℘,ℑ-networkGis S-a,0-EAM total if and only if there is a bijectionδ:VG⟶1,2,…,℘such that the setS=δx+δy|xy∈EGconsists ofℑconsecutive integers. In such a case,δextends to an S-a,0-EAM total labelling ofGwith magic constantℜ=℘+ℑ+s, wheres=minSandS=ℜ−℘+1,ℜ−℘+2,…,ℜ−℘+ℑ.In Lemma1, the sum δx+δy is called as edge sum for each edge xy∈EG. 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-a,0-EAM total, if the edge sums are positive consecutive integers. The following result is also very pertinent as far as S-a,0-EAM total networks are concerned.Theorem 1.
A simple network G admits an S-(a, 0)-EAM total labelling⇔G admits an S-(a-EG+ 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 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]).


### 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 wheelW6, the helm H6, and prism D5 in Figure 1 as a specific example. The edges of the networks W6 and H6 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 D5 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 D5, 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].Figure 1
Vertex-antimagic total labelling of the wheelW6 and helm H6 and edge-magic labelling total of the prism D5 [44,45].
(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.


## 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 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]).


## 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 wheelW6, the helm H6, and prism D5 in Figure 1 as a specific example. The edges of the networks W6 and H6 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 D5 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 D5, 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].Figure 1
Vertex-antimagic total labelling of the wheelW6 and helm H6 and edge-magic labelling total of the prism D5 [44,45].
(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 Section2.1, the S-a,0-EAM total labelling on the union of K2,m∘Cn 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 Cn and pancyclic networks H1 and H2 (planar also). Further, S-a,0-EAM total labelling of planar pancyclic network Γn and symmetric lattice networks Ln1, Ln2, and Ln3 shall be exhibited in Sections 2.3 and 2.4, respectively.


### 2.1. S-a,0-EAM Total Labelling of the Disjoint Union of Cn°K2,m and Pn, Sn, and mP2

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 nK2,m?In fact, the rooted productCn°K2,m contains n copies of the complete bipartite network K2,m. A cycle Cn is a 2-regular network of order n, whereas the complete bipartite network K2,m is class-wise regular in which one partitioned class of vertices is 2-regular and other is m-regular, where VK2,m=m+2.Theorem 2.
For2≤m≡0mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪K1,n∪n−1/2K1admits an S-a,0-EAM total labelling havinga=3mn+8n+2.Proof.
Consider the networkG1≅Cn∘K2,m∪K1,n∪n−1/2K1 with connection scheme as below:(1)VG1=xji:1≤i≤n1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n∪ci:1≤i≤n−1/2∪c,EG1=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪cpi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG1=7n+2mn+1/2 and EG1=2nm+1. Define a labelling f1:VG1⟶1,2,…,7n+2mn+1/2 as(i)
Form=2:(2)f1xji=5n+i2,j=1,i≡1mod2,1≤i≤n,4n+i2,j=1,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=2,1≤i≤n.(ii)
Form>2:(3)f1xji=5n+i2,j=m2,i≡1mod2,1≤i≤n,4n+i2,j=m2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m2+1,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m2−1.nj−i−1+n:1≤i≤n,m2+2≤j≤m.(iii)
The remaining labels form≥2 are as follows:(4)f1yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f1zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1.f1pi=i,1≤i≤n,f1c=7n+2mn+12,f1ci=mn+3n+i,1≤i≤n−12.
All edge sums generated by the labelling schemef1 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. So, by Lemma 1, f1 extends to an S-a,0-EAM total labelling of the network G1 having magic constant a=3mn+8n+2.Theorem 3.
For3≤m≡1mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪K1,n∪n−1/2K1admits an S-a,0-EAM total labelling witha=3mn+8n+2.Proof.
ConsiderG2≅Cn∘K2,m∪K1,n∪n−1/2K1, a network with odd m as follows:(5)VG2=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n∪ci:1≤i≤n−1/2∪c,EG2=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪cpi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG2=7n+2mn+1/2 and EG2=2nm+1. We define a labelling f2:VG2⟶1,2,…,7n+2mn+1/2 as follows:(i)
Form=3:(6)f2xji=5n+i2,j=2,i≡1mod2,1≤i≤n,4n+i2,j=2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=3,1≤i≤n,4n−i−1,j=1,1≤i≤n.(ii)
Form>3:(7)f2xji=5n+i2,j=m+12,i≡1mod2,1≤i≤n,4n+i2,j=m+12,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m+32,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m−12.nj−i−1+n:1≤i≤n,m+52≤j≤m.(iii)
The remaining labels form≥3 are as follows:(8)f2yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f2zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1,f2pi=i,1≤i≤n,f2c=7n+2mn+12,f2ci=mn+3n+i,1≤i≤n−12.
All edge sums generated by the above labelling schemef2 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. By Lemma 1, f2 extends to an S-a,0-EAM total labelling of the network G2 having a=3mn+8n+2.Theorem 4.
For2≤m≡0mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪nP2admits an S-a,0-EAM total labelling witha=6mn+17n+3/2.Proof.
LetG3≅Cn∘K2,m∪nP2 be a network for nn≡1mod2 with vertices:(9)VG3=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi,qi:1≤i≤n,EG3=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪piqi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG3=nm+4 and EG3=2nm+1. A labelling function f3:VG3⟶1,2,…,nm+4 is defined as follows:(i)
Form=2:(10)f3xji=125n+i,j=1,1≤i≤n,i≡1mod2,124n+i,j=1,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=2,1≤i≤n.(ii)
Form>2:(11)f3xji=125n+i,j=m2,i≡1mod2,1≤i≤n,124n+i,j=m2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m2+1,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m2−1.nj−i−1+n:1≤i≤n,m2+2≤j≤m.(iii)
The remaining labels form≥2 are as follows:(12)f3yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f3zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1.f3qi=nm+1+5n−i+22:i≡1mod2,1≤i≤n,nm+1+6n−i+22:i≡0mod2,2≤i≤n−1,f3pi=i,1≤i≤n.
All edge sums generated by the above labelling schemef3 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. So, by Lemma 1, f3 extends to an S-a,0-EAM total labelling of the network G3 admitting magic constant a=6mn+17n+3/2.Theorem 5.
For3≤m≡1mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪nP2admits an S-a,0-EAM total labelling witha=6mn+17n+3/2.Proof.
Consider the networkG4≅Cn∘K2,m∪nP2, for odd m with the construction:(13)VG4=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi,qi:1≤i≤n,EG4=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪piqi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG4=nm+4 and EG4=2nm+1. Define a function f4:VG4⟶1,2,…,nm+4 as follows:(i)
Form=3:(14)f4xji=125n+i,j=2,i≡1mod2,1≤i≤n,124n+i,j=2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=3,1≤i≤n,4n−i−1,j=1,1≤i≤n.(ii)
Form>3:(15)f4xji=125n+i,j=m+12,i≡1mod2,1≤i≤n,124n+i,j=m+12,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m+32,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m−12.nj−i−1+n:1≤i≤n,m+52≤j≤m.(iii)
The remaining labels form≥3 are as follows:(16)f4yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f4zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1,f4qi=nm+1+5n−i+22,i≡1mod2,1≤i≤n,nm+1+6n−i+22,i≡0mod2,2≤i≤n−1,f4pi=i,1≤i≤n.
All edge sums generated by the above labelling schemef4 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. So, by Lemma 1, f4 extends to an S-a,0-EAM total labelling of the network G4 admitting a=6mn+17n+3/2.Theorem 6.
For2≤m≡0mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪Pn+1admits an S-a,0-EAM total labelling witha=6mn+17n+3/2.Proof.
Consider the networkG5≅Cn∘K2,m∪Pn+1, for n≥3 odd, with connections:(17)VG5=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n+1,EG5=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪y1yn∪pipi+1:1≤i≤n∪yiyi+1:1≤i≤n−1.
Here,VG5=3n+mn+1 and EG5=2nm+1. A labelling f5:VG5⟶1,2,…,mn+3n+1 is defined as follows:(i)
Form=2:(18)f5xji=124n+i+1,j=1,i≡1mod2,1≤i≤n,123n+i+1,j=1,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=2,1≤i≤n.(ii)
Form>2:(19)f5xji=124n+i+1,j=m2,i≡1mod2,1≤i≤n,123n+i+1,j=m2,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=m2+1,1≤i≤n,122nj+5n−2i+3,1≤i≤n,1≤j≤m2−1,122nj+n−2i+3:1≤i≤n,m2+2≤j≤m.(iii)
The remaining labels form≥2 are as follows:(20)f5yi=n+i+22,i≡1mod2,1≤i≤n,2n+i+22,i≡0mod2,2≤i≤n−1,f5zi=nm+1+n+i+22,i≡1mod2,1≤i≤n,nm+1+2n+i+22,i≡0mod2,2≤i≤n−1,f5pi=i+12,i≡1mod2,1≤i≤n,122mn+5n+i+1,i≡0mod2,2≤i≤n+1.
All edge sums generated by the above labelling schemef5 constitute a sequence of consecutive integer 3n+5/2,3n+7/2,3n+9/2,…,4mn+7n+3/2. Therefore, by Lemma 1, f5 extends to an S-a,0-EAM total labelling of the network G5 admitting magic constant a=6mn+13n+7/2.Theorem 7.
For3≤m≡1mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪Pn+1admits an S-a,0-EAM total labelling witha=6mn+13n+7/2.Proof.
Consider the networkG6≅Cn∘K2,m∪Pn+1, for both m and n≥3, as follows:(21)VG6=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n+1,EG6=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪y1yn∪pipi+1:1≤i≤n∪yiyi+1:1≤i≤n−1.
Here,VG6=3n+mn+1 and EG6=2nm+1. A labelling f6:VG6⟶1,2,…,3n+mn+1 is designed as follows:(i)
Form=3:(22)f6xji=124n+i+1,j=2,i≡1mod2,1≤i≤n,123n+i+1,j=2,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=3,1≤i≤n,127n−2i+3,j=1,1≤i≤n.(ii)
Form>3:(23)f6xji=124n+i+1,j=m+12,i≡1mod2,1≤i≤n,123n+i+1,j=m+12,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=m+32,1≤i≤n,122nj+5n−2i+3,1≤i≤n,1≤j≤m−12,122nj+n−2i+3,1≤i≤n,m+52≤j≤m.(iii)
The remaining labels form≥3 are as follows:(24)f6yi=n+i+22,i≡1mod2,1≤i≤n,2n+i+22,i≡0mod2,2≤i≤n−1,f6zi=nm+1+n+i+22,i≡1mod2,1≤i≤n,nm+1+2n+i+22,i≡0mod2,2≤i≤n−1,f6pi=i+12,i≡1mod2,1≤i≤n,122mn+5n+i+1,i≡0mod2,2≤i≤n+1.
The edge sums generated by the above labelling schemef6 constitute a sequence of consecutive integer 3n+5/2,3n+7/2,3n+9/2,…,4mn+7n+3/2. By Lemma 1, f6 extends to an S-a,0-EAM total labelling of the network G6 having magic constant a=6mn+13n+7/2.
Observations. The network Cn is S-a,0-EAM total for n≡1mod2 only [8], pointing out that C4 is not S-a,0-EAM total. The networks in Theorems 2, 4, and 6 contain interesting substructures. Keeping m=2 fixed for these results, S-a,0-EAM total families of networks involving n copies of C4 can be obtained. For instance, S-a,0-EAM total labelling (again by using Lemma 1) of C5∘C4∪K1,5∪2K1, C5∘C4∪5P2, and C5∘C4∪P6 is presented in Figure 2.
The following results from Theorems2–7 are direct consequences of Theorem 1.Figure 2
An S-a,0-EAM total labelling (using Lemma 1) of the cyclic networks (a)C5∘C4∪K1,5∪2K1, (b) C5∘C4∪5P2, and (c) C5∘C4∪P6.
(a)(b)(c)Theorem 8.
∀m≥2 and 3≤n≡1mod2,Cn∘K2,m∪K1,n∪n−1/2K1admits an S-mn+6n+3,2-EAM total labelling.Theorem 9.
∀m≥2 and 3≤n≡1mod2,Cn∘K2,m∪nP2admits an S-2mn+13n+5/2,2-EAM total labelling.Theorem 10.
∀m≥2 and 3≤n≡1mod2,Cn∘K2,m∪Pn+1admits an S-2mn+9n+9/2,2-EAM total labelling.


### 2.2. S-a,0-EAM Total Labelling of Rooted Product of Pancyclic Networks with Cn

The 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 Cn.A specific pancyclic networkH1 is defined as follows.Definition 5.
H1is a pancyclic network having the following construction.(25)VH1=y,x1,x2,x3,x4,x5,x6,z,EH1=x1x3,x3x5,x2x4,x4x6,x1x2,x3x4,x5x6,x2x3,x4x5∪yx1,yx2,zx5,zx6.Theorem 11.
Forn≡1mod2, the rooted productCn∘H1admits an S-a,0-EAM total labelling having magic constanta=45n+3/2.Proof.
(i)
Forn=1, C1∘H1≅H1. The vertex labelling y,x1,x2,x3,x4,x5,x6,y,z:2,1,4,3,6,5,8,7 extends to an S-24,0-EAM total labelling of Cn∘H1, by Lemma 1.(ii)
Forn≥3.
Consider the networkCn∘H1 with VCn∘H1=8n and ECn∘H1=14n connected as per the following scheme:(26)VCn∘H1=xi,vi:1≤i≤n∪yi,zi,wi:1≤i≤2n,ECn∘H1=yiyi+1,wiwi+1,zizi+1:1≤i≤2n−1,i≡1mod2∪yizi,ziwi:1≤i≤2n∪xixi+1:1≤i≤n−1∪x1xn∪ziyi+1,wizi+1:1≤i≤2n−1,i≡1mod2∪xiy2i−1,xiy2i,viw2i−1,viw2i:1≤i≤n.
Consider a labellingψ1:VCn∘H1⟶1,2,…,VCn∘H1=8n defined as(27)ψ1xi=i+12,i≡1mod2,1≤i≤n,i+n+12,i≡0mod2,2≤i≤n−1,ψ1yi=6n−i+22,i≡0mod2,2≤i≤2n,6n+i+14,i≡1mod4,1≤i≤2n−1,4n+i+14,i≡3mod4,3≤i≤2n−3,ψ1zi=16n+i4,i≡0mod4,4≤i≤2n−2,12n+i+34,i≡1mod4,1≤i≤2n−1,18n+i4,i≡2mod4,2≤i≤2n,14n+i+34,i≡3mod4,3≤i≤2n−3,ψ1wi=12n−i+12,i≡1mod2,1≤i≤2n−1,26n+i+24,i≡0mod4,4≤i≤2n−2,24n+i+24,i≡2mod4,2≤i≤2n,ψ1vi=15n+i2,i≡1mod2,1≤i≤n,14n+i2,i≡0mod2,2≤i≤n−1.
All edge sums generated by the above labelling schemeψ1 form a sequence of consecutive integer n+3/2,n+5/2,n+7/2,…,29n+1/2. Therefore, by Lemma 1, ψ1 extends to an S-a,0-EAM total labelling of the network Cn∘H1 having magic constant 45n+3/2.Definition 6.
We define a pancyclic networkH2≇H1having vertex set:(28)VH2=y,x1,x2,x3,x4,x5,x6,z,EH2=x1x3,x3x5,x2x4,x4x6,x1x2,x5x6,x2x3,x4x5∪yx1,yx2,zx5,zx6,yz.Theorem 12.
Forn≡1mod2, the rooted productCn∘H2admits an S-a,0-EAM total labelling with magic constanta=45n+3/2.Proof.
(i)
Forn=1, C1∘H2≅H2.(ii)
Forn≥3.
ConsiderCn∘H2 with VCn∘H2=8n and ECn∘H2=14n with the following connection:(29)VCn∘H2=xi,vi:1≤i≤n∪wi,yi,zi:1≤i≤2n,EPn∘H2=yiyi+1,wiwi+1:1≤i≤2n−1,i≡1mod2∪yizi,ziwi:1≤i≤2n∪xixi+1:1≤i≤n−1∪x1xn∪ziyi+1,wizi+1:1≤i≤2n−1,i≡1mod2∪xiy2i−1,xiy2i,viw2i−1,viw2i,xivi:1≤i≤n.
The labelling scheme forn=1 and n≥3 is the same as ψ1 designed in Theorem 11.
A direct derivation from Theorem1 is given as follows.Theorem 13.
Forn≡1mod2,Cn∘H1andCn∘H2are S-17n+5/2,2-EAM total.


### 2.3. S-a,0-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-a,0-EAM total labelling of a pancyclic class of networks involving chains of cycle C4. Here, we shall introduce a pancyclic family of networks involving chains of cycle C6, while our point of convergence is the S-a,0-EAM total labelling of this class. Thus, we further extend the results of Baig et al. [29].Definition 7.
The pancyclic networkΓnis a network with orderVΓn=6nandEΓn=12n−3, with structure as follows:(30)VΓn=zi,xi,yi:1≤i≤2n,EΓn=yiyi+1,xixi+1,zizi+1:1≤i≤2n−1∪xiyi,xizi,yizi:1≤i≤2n.
Figure3 reveals general formation of Γn.
In Figure4, we have shown the network Γ2 and its contained cycles of orders 3,4,…,12.
In the upcoming result, we show that the pancyclic networkΓn is S-a,0-EAM total.Figure 3
The general formation of the pancyclic networkΓn.Figure 4
The networkΓ2 having order 12 and its contained cycles C3,C4,…,C12.Theorem 14.
For all positive integersn, the pancyclic networkΓnis S-a,0-EAM total having magic constant18n.Proof.
(i)
Forn=1, the labelling z1,z2,x1,x2,y1,y2:3,5,2,4,1,6 extends to an S-18,0-EAM total labelling of Γ1 by Lemma 1.(ii)
Forn≥3.
Define here a labellingg:VΓn⟶1,2,…,6n as(31)gxi=3i:i≡1mod2,1≤i≤2n−1,3i−1:i≡0mod2,2≤i≤2n,gyi=3i−2:i≡1mod2,1≤i≤2n−1,3i:i≡0mod2,2≤i≤2n,gzi=3i−1:i≡1mod2,1≤i≤2n−1,3i−2:i≡0mod2,2≤i≤2n.
All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers3,4,…,12n−1. Therefore, by Lemma 1, g extends to an S-a,0-EAM total labelling of Γn having magic constant a=18n.
Again from Theorem1, we have a direct consequence as follows.Theorem 15.
For all positive integersn, the pancyclic networkΓnadmits an S-6n+4,2-EAM total labelling.


### 2.4. S-a,0-EAM Total Labelling of Symmetric Lattice Networks

Definition 8.
Consider a tripartite networkThaving vertex-edge connections as follows:(32)VT=x,y,z,u,v,w,ET=uv,vw,xy,yz,uy,wy,xv,zv,vy.
In this section, we study the S-a,0-EAM total labelling of symmetric lattice networks Ln1, Ln2, and Ln3. These symmetric lattices contain n copies of the tripartite network T.Definition 9.
(i)
Forn=1, L11≅T.(ii)
Forn≥2.
The lattice networkLn1is a network with order6nand size12n−3defined as follows:(33)VLn1=xi,yi,ci:1≤i≤2n.ELn1=ciyi+1,cixi+1,ciyi,cixi:1≤i≤2n−1,i≡1mod2∪cici+1:2≤i≤2n−1∪xixi+1,yiyi+1:i≡0mod2,2≤i≤2n−1,∪ciyi−1,cixi−1,ciyi,cixi:i≡0mod2,2≤i≤2n.
Figure5 illustrates the general formation of the lattice network Ln1, ∀n∈ℕ.Figure 5
The general formation of the lattice networkLn1,∀n∈ℕ.Theorem 16.
For all positive integersn, the lattice networkLn1is S-a,0-EAM total having magic constant18n.Proof.
(i)
Forn=1: the vertex labelling x1,c1,y1,x2,c2,y2:3,1,2,5,6,4 extends to an S-18,0-EAM total labelling of L11, by Lemma 1.(ii)
Forn>1.
We are defining a labellingφ1:VLn1⟶1,2,…,6n as follows:(34)φ1xi=3i:i≡1mod2,1≤i≤2n−1,3i−1:i≡0mod2,2≤i≤2n,φ1yi=3i−1:i≡1mod2,1≤i≤2n−1,3i−2:i≡0mod2,2≤i≤2n,φ1ci=3i−2:i≡1mod2,1≤i≤2n−1,3i:i≡0mod2,2≤i≤2n.
All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers3,4,…,12n−3. So, by Lemma 1, φ1 extends to an S-a,0-EAM total labelling of Ln1 admitting magic constant 18n.Definition 10.
(i)
Forn=1, L12≅T.(ii)
Forn≥2.
The lattice networkLn2having order10n−4and size20n−11is defined as follows:(35)VLn2=xi,yi:1≤i≤2n∪ci:1≤i≤22n−1∪ui,vi:1≤i≤n−1.ELn2=ciyi+2/2,cixi+2/2,ciyi/2,cixi/2:2≤i≤22n−1,i≡2mod4∪ciyi+2/2,ciyi+3/2,cixi+2/2,cixi+3/2:i≡1mod4,1≤i≤4n−3∪cici+1:1≤i≤4n−3,i≡1mod4∪uivi:1≤i≤n−1∪xiui/2,yivi/2:2≤i≤2n−1,i≡0mod2∪uix2i+1,viy2i+1,uic4i−1,vic4i−1,uic4i,vic4i:1≤i≤n−1∪cici+1:i≡0mod2,2≤i≤4n−1.
In Figure6, we have presented the general formation of the lattice network Ln2, ∀n∈ℕ.Figure 6
The general formation of the lattice networkLn2,∀n∈ℕ.Theorem 17.
For all positive integersn, the lattice networkLn2is S-a,0-EAM total with magic constant30n−12.Proof.
(i)
Forn=1: the vertex labelling x1,c1,y1,x2,c2,y2:3,1,2,5,6,4 extends to an S-18,0-EAM total labelling of L12, by Lemma 1.(ii)
Forn>1.
A labellingφ2:VLn2⟶1,2,…,10n−4 is defined as follows:(36)φ2xi=5i−2:i≡1mod2,1≤i≤2n−1,5i−5:i≡0mod2,2≤i≤2n,φ2yi=5i−3:i≡1mod2,1≤i≤2n−1,5i−6:i≡0mod2,2≤i≤2n,φ2ci=125i−3:1≤i≤4n−3,i≡1mod4,125i+2:2≤i≤4n−2,i≡2mod4,125i−1:3≤i≤4n−5,i≡3mod4,125i:4≤i≤4n−1,i≡0mod4,φ2ui=10i−1,1≤i≤n−1,φ2vi=10i−2,1≤i≤n−1.
All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers3,4,…,20n−9. So, by Lemma 1, φ2 extends to an S-a,0-EAM total labelling of Ln2 having magic constant 30n−12.Definition 11.
(i)
Forn=1, L13≅T.(ii)
Forn≥2.
The lattice networkLn3having order10n−4and size20n−11is defined as follows:(37)VLn3=xi,yi:1≤i≤2n∪ci:1≤i≤22n−1∪ui,vi:1≤i≤n−1,ELn3=ciyi+2/2,cixi+2/2,ciyi/2,cixi/2:2≤i≤22n−1,i≡2mod4∪ciyi+2/2,ciyi+3/2,cixi+1/2,cixi+3/2:1≤i≤4n−3,i≡1mod4∪xiui/2,yivi/2:i≡0mod2,2≤i≤2n−1∪cici+1:1≤i≤4n−3∪uix2i+1,viy2i+1,uic4i−1,vic4i−1,uic4i,vic4i:1≤i≤n−1.
Figure7 illustrates the general form of the lattice network Ln3, ∀n∈ℕ.Figure 7
The general formation of the lattice networkLn3,∀n∈ℕ.Theorem 18.
For all positive integersn, the lattice networkLn3is S-a,0-EAM total with magic constant30n−12.Proof.
ForLn3, the labelling design is similar as in Theorem 17.


## 2.1. S-a,0-EAM Total Labelling of the Disjoint Union of Cn°K2,m and Pn, Sn, and mP2

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 nK2,m?In fact, the rooted productCn°K2,m contains n copies of the complete bipartite network K2,m. A cycle Cn is a 2-regular network of order n, whereas the complete bipartite network K2,m is class-wise regular in which one partitioned class of vertices is 2-regular and other is m-regular, where VK2,m=m+2.Theorem 2.
For2≤m≡0mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪K1,n∪n−1/2K1admits an S-a,0-EAM total labelling havinga=3mn+8n+2.Proof.
Consider the networkG1≅Cn∘K2,m∪K1,n∪n−1/2K1 with connection scheme as below:(1)VG1=xji:1≤i≤n1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n∪ci:1≤i≤n−1/2∪c,EG1=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪cpi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG1=7n+2mn+1/2 and EG1=2nm+1. Define a labelling f1:VG1⟶1,2,…,7n+2mn+1/2 as(i)
Form=2:(2)f1xji=5n+i2,j=1,i≡1mod2,1≤i≤n,4n+i2,j=1,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=2,1≤i≤n.(ii)
Form>2:(3)f1xji=5n+i2,j=m2,i≡1mod2,1≤i≤n,4n+i2,j=m2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m2+1,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m2−1.nj−i−1+n:1≤i≤n,m2+2≤j≤m.(iii)
The remaining labels form≥2 are as follows:(4)f1yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f1zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1.f1pi=i,1≤i≤n,f1c=7n+2mn+12,f1ci=mn+3n+i,1≤i≤n−12.
All edge sums generated by the labelling schemef1 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. So, by Lemma 1, f1 extends to an S-a,0-EAM total labelling of the network G1 having magic constant a=3mn+8n+2.Theorem 3.
For3≤m≡1mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪K1,n∪n−1/2K1admits an S-a,0-EAM total labelling witha=3mn+8n+2.Proof.
ConsiderG2≅Cn∘K2,m∪K1,n∪n−1/2K1, a network with odd m as follows:(5)VG2=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n∪ci:1≤i≤n−1/2∪c,EG2=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪cpi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG2=7n+2mn+1/2 and EG2=2nm+1. We define a labelling f2:VG2⟶1,2,…,7n+2mn+1/2 as follows:(i)
Form=3:(6)f2xji=5n+i2,j=2,i≡1mod2,1≤i≤n,4n+i2,j=2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=3,1≤i≤n,4n−i−1,j=1,1≤i≤n.(ii)
Form>3:(7)f2xji=5n+i2,j=m+12,i≡1mod2,1≤i≤n,4n+i2,j=m+12,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m+32,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m−12.nj−i−1+n:1≤i≤n,m+52≤j≤m.(iii)
The remaining labels form≥3 are as follows:(8)f2yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f2zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1,f2pi=i,1≤i≤n,f2c=7n+2mn+12,f2ci=mn+3n+i,1≤i≤n−12.
All edge sums generated by the above labelling schemef2 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. By Lemma 1, f2 extends to an S-a,0-EAM total labelling of the network G2 having a=3mn+8n+2.Theorem 4.
For2≤m≡0mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪nP2admits an S-a,0-EAM total labelling witha=6mn+17n+3/2.Proof.
LetG3≅Cn∘K2,m∪nP2 be a network for nn≡1mod2 with vertices:(9)VG3=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi,qi:1≤i≤n,EG3=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪piqi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG3=nm+4 and EG3=2nm+1. A labelling function f3:VG3⟶1,2,…,nm+4 is defined as follows:(i)
Form=2:(10)f3xji=125n+i,j=1,1≤i≤n,i≡1mod2,124n+i,j=1,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=2,1≤i≤n.(ii)
Form>2:(11)f3xji=125n+i,j=m2,i≡1mod2,1≤i≤n,124n+i,j=m2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m2+1,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m2−1.nj−i−1+n:1≤i≤n,m2+2≤j≤m.(iii)
The remaining labels form≥2 are as follows:(12)f3yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f3zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1.f3qi=nm+1+5n−i+22:i≡1mod2,1≤i≤n,nm+1+6n−i+22:i≡0mod2,2≤i≤n−1,f3pi=i,1≤i≤n.
All edge sums generated by the above labelling schemef3 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. So, by Lemma 1, f3 extends to an S-a,0-EAM total labelling of the network G3 admitting magic constant a=6mn+17n+3/2.Theorem 5.
For3≤m≡1mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪nP2admits an S-a,0-EAM total labelling witha=6mn+17n+3/2.Proof.
Consider the networkG4≅Cn∘K2,m∪nP2, for odd m with the construction:(13)VG4=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi,qi:1≤i≤n,EG4=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪piqi:1≤i≤n∪yiyi+1:1≤i≤n−1∪y1yn.
Here,VG4=nm+4 and EG4=2nm+1. Define a function f4:VG4⟶1,2,…,nm+4 as follows:(i)
Form=3:(14)f4xji=125n+i,j=2,i≡1mod2,1≤i≤n,124n+i,j=2,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=3,1≤i≤n,4n−i−1,j=1,1≤i≤n.(ii)
Form>3:(15)f4xji=125n+i,j=m+12,i≡1mod2,1≤i≤n,124n+i,j=m+12,i≡0mod2,2≤i≤n−1,nm+2−i−1,j=m+32,1≤i≤n,nj+3−i−1,1≤i≤n,1≤j≤m−12.nj−i−1+n:1≤i≤n,m+52≤j≤m.(iii)
The remaining labels form≥3 are as follows:(16)f4yi=i+2n+12,i≡1mod2,1≤i≤n,i+3n+12,i≡0mod2,2≤i≤n−1,f4zi=nm+1+i+2n+12,i≡1mod2,1≤i≤n,nm+1+i+3n+12,i≡0mod2,2≤i≤n−1,f4qi=nm+1+5n−i+22,i≡1mod2,1≤i≤n,nm+1+6n−i+22,i≡0mod2,2≤i≤n−1,f4pi=i,1≤i≤n.
All edge sums generated by the above labelling schemef4 constitute a sequence of consecutive integer 5n+3/2,5n+5/2,5n+7/2,…,4mn+9n+1/2. So, by Lemma 1, f4 extends to an S-a,0-EAM total labelling of the network G4 admitting a=6mn+17n+3/2.Theorem 6.
For2≤m≡0mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪Pn+1admits an S-a,0-EAM total labelling witha=6mn+17n+3/2.Proof.
Consider the networkG5≅Cn∘K2,m∪Pn+1, for n≥3 odd, with connections:(17)VG5=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n+1,EG5=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪y1yn∪pipi+1:1≤i≤n∪yiyi+1:1≤i≤n−1.
Here,VG5=3n+mn+1 and EG5=2nm+1. A labelling f5:VG5⟶1,2,…,mn+3n+1 is defined as follows:(i)
Form=2:(18)f5xji=124n+i+1,j=1,i≡1mod2,1≤i≤n,123n+i+1,j=1,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=2,1≤i≤n.(ii)
Form>2:(19)f5xji=124n+i+1,j=m2,i≡1mod2,1≤i≤n,123n+i+1,j=m2,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=m2+1,1≤i≤n,122nj+5n−2i+3,1≤i≤n,1≤j≤m2−1,122nj+n−2i+3:1≤i≤n,m2+2≤j≤m.(iii)
The remaining labels form≥2 are as follows:(20)f5yi=n+i+22,i≡1mod2,1≤i≤n,2n+i+22,i≡0mod2,2≤i≤n−1,f5zi=nm+1+n+i+22,i≡1mod2,1≤i≤n,nm+1+2n+i+22,i≡0mod2,2≤i≤n−1,f5pi=i+12,i≡1mod2,1≤i≤n,122mn+5n+i+1,i≡0mod2,2≤i≤n+1.
All edge sums generated by the above labelling schemef5 constitute a sequence of consecutive integer 3n+5/2,3n+7/2,3n+9/2,…,4mn+7n+3/2. Therefore, by Lemma 1, f5 extends to an S-a,0-EAM total labelling of the network G5 admitting magic constant a=6mn+13n+7/2.Theorem 7.
For3≤m≡1mod2 and 3≤n≡1mod2, the networkCn∘K2,m∪Pn+1admits an S-a,0-EAM total labelling witha=6mn+13n+7/2.Proof.
Consider the networkG6≅Cn∘K2,m∪Pn+1, for both m and n≥3, as follows:(21)VG6=xji:1≤i≤n,1≤j≤m∪yi,zi:1≤i≤n∪pi:1≤i≤n+1,EG6=yixji:1≤i≤n,1≤j≤m∪zixji:1≤i≤n,1≤j≤m∪y1yn∪pipi+1:1≤i≤n∪yiyi+1:1≤i≤n−1.
Here,VG6=3n+mn+1 and EG6=2nm+1. A labelling f6:VG6⟶1,2,…,3n+mn+1 is designed as follows:(i)
Form=3:(22)f6xji=124n+i+1,j=2,i≡1mod2,1≤i≤n,123n+i+1,j=2,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=3,1≤i≤n,127n−2i+3,j=1,1≤i≤n.(ii)
Form>3:(23)f6xji=124n+i+1,j=m+12,i≡1mod2,1≤i≤n,123n+i+1,j=m+12,i≡0mod2,2≤i≤n−1,122mn+3n−2i+3,j=m+32,1≤i≤n,122nj+5n−2i+3,1≤i≤n,1≤j≤m−12,122nj+n−2i+3,1≤i≤n,m+52≤j≤m.(iii)
The remaining labels form≥3 are as follows:(24)f6yi=n+i+22,i≡1mod2,1≤i≤n,2n+i+22,i≡0mod2,2≤i≤n−1,f6zi=nm+1+n+i+22,i≡1mod2,1≤i≤n,nm+1+2n+i+22,i≡0mod2,2≤i≤n−1,f6pi=i+12,i≡1mod2,1≤i≤n,122mn+5n+i+1,i≡0mod2,2≤i≤n+1.
The edge sums generated by the above labelling schemef6 constitute a sequence of consecutive integer 3n+5/2,3n+7/2,3n+9/2,…,4mn+7n+3/2. By Lemma 1, f6 extends to an S-a,0-EAM total labelling of the network G6 having magic constant a=6mn+13n+7/2.
Observations. The network Cn is S-a,0-EAM total for n≡1mod2 only [8], pointing out that C4 is not S-a,0-EAM total. The networks in Theorems 2, 4, and 6 contain interesting substructures. Keeping m=2 fixed for these results, S-a,0-EAM total families of networks involving n copies of C4 can be obtained. For instance, S-a,0-EAM total labelling (again by using Lemma 1) of C5∘C4∪K1,5∪2K1, C5∘C4∪5P2, and C5∘C4∪P6 is presented in Figure 2.
The following results from Theorems2–7 are direct consequences of Theorem 1.Figure 2
An S-a,0-EAM total labelling (using Lemma 1) of the cyclic networks (a)C5∘C4∪K1,5∪2K1, (b) C5∘C4∪5P2, and (c) C5∘C4∪P6.
(a)(b)(c)Theorem 8.
∀m≥2 and 3≤n≡1mod2,Cn∘K2,m∪K1,n∪n−1/2K1admits an S-mn+6n+3,2-EAM total labelling.Theorem 9.
∀m≥2 and 3≤n≡1mod2,Cn∘K2,m∪nP2admits an S-2mn+13n+5/2,2-EAM total labelling.Theorem 10.
∀m≥2 and 3≤n≡1mod2,Cn∘K2,m∪Pn+1admits an S-2mn+9n+9/2,2-EAM total labelling.


## 2.2. S-a,0-EAM Total Labelling of Rooted Product of Pancyclic Networks with Cn

The 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 Cn.A specific pancyclic networkH1 is defined as follows.Definition 5.
H1is a pancyclic network having the following construction.(25)VH1=y,x1,x2,x3,x4,x5,x6,z,EH1=x1x3,x3x5,x2x4,x4x6,x1x2,x3x4,x5x6,x2x3,x4x5∪yx1,yx2,zx5,zx6.Theorem 11.
Forn≡1mod2, the rooted productCn∘H1admits an S-a,0-EAM total labelling having magic constanta=45n+3/2.Proof.
(i)
Forn=1, C1∘H1≅H1. The vertex labelling y,x1,x2,x3,x4,x5,x6,y,z:2,1,4,3,6,5,8,7 extends to an S-24,0-EAM total labelling of Cn∘H1, by Lemma 1.(ii)
Forn≥3.
Consider the networkCn∘H1 with VCn∘H1=8n and ECn∘H1=14n connected as per the following scheme:(26)VCn∘H1=xi,vi:1≤i≤n∪yi,zi,wi:1≤i≤2n,ECn∘H1=yiyi+1,wiwi+1,zizi+1:1≤i≤2n−1,i≡1mod2∪yizi,ziwi:1≤i≤2n∪xixi+1:1≤i≤n−1∪x1xn∪ziyi+1,wizi+1:1≤i≤2n−1,i≡1mod2∪xiy2i−1,xiy2i,viw2i−1,viw2i:1≤i≤n.
Consider a labellingψ1:VCn∘H1⟶1,2,…,VCn∘H1=8n defined as(27)ψ1xi=i+12,i≡1mod2,1≤i≤n,i+n+12,i≡0mod2,2≤i≤n−1,ψ1yi=6n−i+22,i≡0mod2,2≤i≤2n,6n+i+14,i≡1mod4,1≤i≤2n−1,4n+i+14,i≡3mod4,3≤i≤2n−3,ψ1zi=16n+i4,i≡0mod4,4≤i≤2n−2,12n+i+34,i≡1mod4,1≤i≤2n−1,18n+i4,i≡2mod4,2≤i≤2n,14n+i+34,i≡3mod4,3≤i≤2n−3,ψ1wi=12n−i+12,i≡1mod2,1≤i≤2n−1,26n+i+24,i≡0mod4,4≤i≤2n−2,24n+i+24,i≡2mod4,2≤i≤2n,ψ1vi=15n+i2,i≡1mod2,1≤i≤n,14n+i2,i≡0mod2,2≤i≤n−1.
All edge sums generated by the above labelling schemeψ1 form a sequence of consecutive integer n+3/2,n+5/2,n+7/2,…,29n+1/2. Therefore, by Lemma 1, ψ1 extends to an S-a,0-EAM total labelling of the network Cn∘H1 having magic constant 45n+3/2.Definition 6.
We define a pancyclic networkH2≇H1having vertex set:(28)VH2=y,x1,x2,x3,x4,x5,x6,z,EH2=x1x3,x3x5,x2x4,x4x6,x1x2,x5x6,x2x3,x4x5∪yx1,yx2,zx5,zx6,yz.Theorem 12.
Forn≡1mod2, the rooted productCn∘H2admits an S-a,0-EAM total labelling with magic constanta=45n+3/2.Proof.
(i)
Forn=1, C1∘H2≅H2.(ii)
Forn≥3.
ConsiderCn∘H2 with VCn∘H2=8n and ECn∘H2=14n with the following connection:(29)VCn∘H2=xi,vi:1≤i≤n∪wi,yi,zi:1≤i≤2n,EPn∘H2=yiyi+1,wiwi+1:1≤i≤2n−1,i≡1mod2∪yizi,ziwi:1≤i≤2n∪xixi+1:1≤i≤n−1∪x1xn∪ziyi+1,wizi+1:1≤i≤2n−1,i≡1mod2∪xiy2i−1,xiy2i,viw2i−1,viw2i,xivi:1≤i≤n.
The labelling scheme forn=1 and n≥3 is the same as ψ1 designed in Theorem 11.
A direct derivation from Theorem1 is given as follows.Theorem 13.
Forn≡1mod2,Cn∘H1andCn∘H2are S-17n+5/2,2-EAM total.


## 2.3. S-a,0-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-a,0-EAM total labelling of a pancyclic class of networks involving chains of cycle C4. Here, we shall introduce a pancyclic family of networks involving chains of cycle C6, while our point of convergence is the S-a,0-EAM total labelling of this class. Thus, we further extend the results of Baig et al. [29].Definition 7.
The pancyclic networkΓnis a network with orderVΓn=6nandEΓn=12n−3, with structure as follows:(30)VΓn=zi,xi,yi:1≤i≤2n,EΓn=yiyi+1,xixi+1,zizi+1:1≤i≤2n−1∪xiyi,xizi,yizi:1≤i≤2n.
Figure3 reveals general formation of Γn.
In Figure4, we have shown the network Γ2 and its contained cycles of orders 3,4,…,12.
In the upcoming result, we show that the pancyclic networkΓn is S-a,0-EAM total.Figure 3
The general formation of the pancyclic networkΓn.Figure 4
The networkΓ2 having order 12 and its contained cycles C3,C4,…,C12.Theorem 14.
For all positive integersn, the pancyclic networkΓnis S-a,0-EAM total having magic constant18n.Proof.
(i)
Forn=1, the labelling z1,z2,x1,x2,y1,y2:3,5,2,4,1,6 extends to an S-18,0-EAM total labelling of Γ1 by Lemma 1.(ii)
Forn≥3.
Define here a labellingg:VΓn⟶1,2,…,6n as(31)gxi=3i:i≡1mod2,1≤i≤2n−1,3i−1:i≡0mod2,2≤i≤2n,gyi=3i−2:i≡1mod2,1≤i≤2n−1,3i:i≡0mod2,2≤i≤2n,gzi=3i−1:i≡1mod2,1≤i≤2n−1,3i−2:i≡0mod2,2≤i≤2n.
All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers3,4,…,12n−1. Therefore, by Lemma 1, g extends to an S-a,0-EAM total labelling of Γn having magic constant a=18n.
Again from Theorem1, we have a direct consequence as follows.Theorem 15.
For all positive integersn, the pancyclic networkΓnadmits an S-6n+4,2-EAM total labelling.


## 2.4. S-a,0-EAM Total Labelling of Symmetric Lattice Networks

Definition 8.
Consider a tripartite networkThaving vertex-edge connections as follows:(32)VT=x,y,z,u,v,w,ET=uv,vw,xy,yz,uy,wy,xv,zv,vy.
In this section, we study the S-a,0-EAM total labelling of symmetric lattice networks Ln1, Ln2, and Ln3. These symmetric lattices contain n copies of the tripartite network T.Definition 9.
(i)
Forn=1, L11≅T.(ii)
Forn≥2.
The lattice networkLn1is a network with order6nand size12n−3defined as follows:(33)VLn1=xi,yi,ci:1≤i≤2n.ELn1=ciyi+1,cixi+1,ciyi,cixi:1≤i≤2n−1,i≡1mod2∪cici+1:2≤i≤2n−1∪xixi+1,yiyi+1:i≡0mod2,2≤i≤2n−1,∪ciyi−1,cixi−1,ciyi,cixi:i≡0mod2,2≤i≤2n.
Figure5 illustrates the general formation of the lattice network Ln1, ∀n∈ℕ.Figure 5
The general formation of the lattice networkLn1,∀n∈ℕ.Theorem 16.
For all positive integersn, the lattice networkLn1is S-a,0-EAM total having magic constant18n.Proof.
(i)
Forn=1: the vertex labelling x1,c1,y1,x2,c2,y2:3,1,2,5,6,4 extends to an S-18,0-EAM total labelling of L11, by Lemma 1.(ii)
Forn>1.
We are defining a labellingφ1:VLn1⟶1,2,…,6n as follows:(34)φ1xi=3i:i≡1mod2,1≤i≤2n−1,3i−1:i≡0mod2,2≤i≤2n,φ1yi=3i−1:i≡1mod2,1≤i≤2n−1,3i−2:i≡0mod2,2≤i≤2n,φ1ci=3i−2:i≡1mod2,1≤i≤2n−1,3i:i≡0mod2,2≤i≤2n.
All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers3,4,…,12n−3. So, by Lemma 1, φ1 extends to an S-a,0-EAM total labelling of Ln1 admitting magic constant 18n.Definition 10.
(i)
Forn=1, L12≅T.(ii)
Forn≥2.
The lattice networkLn2having order10n−4and size20n−11is defined as follows:(35)VLn2=xi,yi:1≤i≤2n∪ci:1≤i≤22n−1∪ui,vi:1≤i≤n−1.ELn2=ciyi+2/2,cixi+2/2,ciyi/2,cixi/2:2≤i≤22n−1,i≡2mod4∪ciyi+2/2,ciyi+3/2,cixi+2/2,cixi+3/2:i≡1mod4,1≤i≤4n−3∪cici+1:1≤i≤4n−3,i≡1mod4∪uivi:1≤i≤n−1∪xiui/2,yivi/2:2≤i≤2n−1,i≡0mod2∪uix2i+1,viy2i+1,uic4i−1,vic4i−1,uic4i,vic4i:1≤i≤n−1∪cici+1:i≡0mod2,2≤i≤4n−1.
In Figure6, we have presented the general formation of the lattice network Ln2, ∀n∈ℕ.Figure 6
The general formation of the lattice networkLn2,∀n∈ℕ.Theorem 17.
For all positive integersn, the lattice networkLn2is S-a,0-EAM total with magic constant30n−12.Proof.
(i)
Forn=1: the vertex labelling x1,c1,y1,x2,c2,y2:3,1,2,5,6,4 extends to an S-18,0-EAM total labelling of L12, by Lemma 1.(ii)
Forn>1.
A labellingφ2:VLn2⟶1,2,…,10n−4 is defined as follows:(36)φ2xi=5i−2:i≡1mod2,1≤i≤2n−1,5i−5:i≡0mod2,2≤i≤2n,φ2yi=5i−3:i≡1mod2,1≤i≤2n−1,5i−6:i≡0mod2,2≤i≤2n,φ2ci=125i−3:1≤i≤4n−3,i≡1mod4,125i+2:2≤i≤4n−2,i≡2mod4,125i−1:3≤i≤4n−5,i≡3mod4,125i:4≤i≤4n−1,i≡0mod4,φ2ui=10i−1,1≤i≤n−1,φ2vi=10i−2,1≤i≤n−1.
All edge sums generated by the above labelling scheme constitute a sequence of consecutive integers3,4,…,20n−9. So, by Lemma 1, φ2 extends to an S-a,0-EAM total labelling of Ln2 having magic constant 30n−12.Definition 11.
(i)
Forn=1, L13≅T.(ii)
Forn≥2.
The lattice networkLn3having order10n−4and size20n−11is defined as follows:(37)VLn3=xi,yi:1≤i≤2n∪ci:1≤i≤22n−1∪ui,vi:1≤i≤n−1,ELn3=ciyi+2/2,cixi+2/2,ciyi/2,cixi/2:2≤i≤22n−1,i≡2mod4∪ciyi+2/2,ciyi+3/2,cixi+1/2,cixi+3/2:1≤i≤4n−3,i≡1mod4∪xiui/2,yivi/2:i≡0mod2,2≤i≤2n−1∪cici+1:1≤i≤4n−3∪uix2i+1,viy2i+1,uic4i−1,vic4i−1,uic4i,vic4i:1≤i≤n−1.
Figure7 illustrates the general form of the lattice network Ln3, ∀n∈ℕ.Figure 7
The general formation of the lattice networkLn3,∀n∈ℕ.Theorem 18.
For all positive integersn, the lattice networkLn3is S-a,0-EAM total with magic constant30n−12.Proof.
ForLn3, the labelling design is similar as in Theorem 17.


## 3. Illustration through Examples and Proposed Open Problems

### 3.1. Examples

The S-132,0-EAM total labelling of C5∘K2,6∪K1,5∪2K1 and S-163,0-EAM total labelling of C7∘K2,5∪K1,7∪3K1 are presented, respectively, in Figures 8(a) and 8(b). In Figure 8(a), the parameters are {n=5, m=6} and a=132, while in Figure 8(b), the parameters are {n=7, m=5} and a=163. These are perfect according to our depiction of the magic constants in the proofs of Theorems 2 and 3.Figure 8
An S-132,0-EAM total labelling of C5∘K2,6∪K1,5∪2K1 (←A) and an S-163,0-EAM total labelling of C7∘K2,5∪K1,7∪3K1 (←B).
(a)(b)Similarly, Figures9 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 a here are perfectly similar as depicted in our main findings.Figure 9
An S-134,0-EAM total labelling of C5∘K2,6∪5P2 (⟶A) and an S-165,0-EAM total labelling of C7∘K2,5∪5P2 (⟶B).
(a)(b)Figure 10
An S-126,0-EAM total labelling of C5∘K2,6∪P6 (⟶A) and an S-154,0-EAM total labelling of C7∘K2,5∪P8 (⟶B).
(a)(b)Figures11(a) and 11(b) illustrate Theorems 11 and 12, respectively, for n=5.Figure 11
An S-114,0-EAM total labelling of C5∘H1⟶A and C5∘H2⟶B.
(a)(b)Figure12 reveals an example of Theorem 15 corresponding to parameter n=3.Figure 12
An S-54,0-EAM total labelling of Γ3.Figures13–15 refer to the illustration of S-a,0-EAM total labelling of lattice networks L51, L42, and L43 (Theorems 16–18).Figure 13
An S-90,0-EAM total labelling of the lattice network L51.Figure 14
An S-108,0-EAM total labelling of the lattice network L42.Figure 15
An S-108,0-EAM total labelling of the lattice network L43.Due to facilitation of Lemma1, 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.


### 3.2. Open Problems

The open problems relevant to the findings (Theorems2–7) of Section 2 are proposed as follows:(i)
For2≤n≡0mod2, obtain any S-a,0-EAM total labelling of the Cn∘K2,m∪K1,n∪n−1/2K1.(ii)
For2≤n≡0mod2, obtain any S-a,0-EAM total labelling of Cn∘K2,m∪nP2.(iii)
For2≤n≡0mod2, obtain any S-a,0-EAM total labelling of Cn∘K2,m∪Pn+1.(iv)
For2≤n≡0mod2, determine S-a,0-EAM total labelling of G1, G2, G3, G4, G5, and G6 for any other magic constants (i.e., for any other value of a) than computed here.(v)
Forl,m and n positive integers, determine any S-a,0-EAM total labelling for the following networks:(1)
Cn∘K2,m∪K1,ℓ(2)
Cn∘K2,m∪lP2(3)
Cn∘K2,m∪PlOpen problems related to Theorems11 and 12 are given as follows:(i)
For2≤n≡0mod2, find any S-a,0-EAM total labelling of Cn∘H1.(ii)
For2≤n≡0mod2, find any S-a,0-EAM total labelling of Cn∘H2.(iii)
For3≤n≡1mod2, determine some S-a,0-EAM total labelling of Cn∘H1 and Cn∘H2 with a different magic constant than obtained here, i.e., for any other value of a.


## 3.1. Examples

The S-132,0-EAM total labelling of C5∘K2,6∪K1,5∪2K1 and S-163,0-EAM total labelling of C7∘K2,5∪K1,7∪3K1 are presented, respectively, in Figures 8(a) and 8(b). In Figure 8(a), the parameters are {n=5, m=6} and a=132, while in Figure 8(b), the parameters are {n=7, m=5} and a=163. These are perfect according to our depiction of the magic constants in the proofs of Theorems 2 and 3.Figure 8
An S-132,0-EAM total labelling of C5∘K2,6∪K1,5∪2K1 (←A) and an S-163,0-EAM total labelling of C7∘K2,5∪K1,7∪3K1 (←B).
(a)(b)Similarly, Figures9 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 a here are perfectly similar as depicted in our main findings.Figure 9
An S-134,0-EAM total labelling of C5∘K2,6∪5P2 (⟶A) and an S-165,0-EAM total labelling of C7∘K2,5∪5P2 (⟶B).
(a)(b)Figure 10
An S-126,0-EAM total labelling of C5∘K2,6∪P6 (⟶A) and an S-154,0-EAM total labelling of C7∘K2,5∪P8 (⟶B).
(a)(b)Figures11(a) and 11(b) illustrate Theorems 11 and 12, respectively, for n=5.Figure 11
An S-114,0-EAM total labelling of C5∘H1⟶A and C5∘H2⟶B.
(a)(b)Figure12 reveals an example of Theorem 15 corresponding to parameter n=3.Figure 12
An S-54,0-EAM total labelling of Γ3.Figures13–15 refer to the illustration of S-a,0-EAM total labelling of lattice networks L51, L42, and L43 (Theorems 16–18).Figure 13
An S-90,0-EAM total labelling of the lattice network L51.Figure 14
An S-108,0-EAM total labelling of the lattice network L42.Figure 15
An S-108,0-EAM total labelling of the lattice network L43.Due to facilitation of Lemma1, 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.


## 3.2. Open Problems

The open problems relevant to the findings (Theorems2–7) of Section 2 are proposed as follows:(i)
For2≤n≡0mod2, obtain any S-a,0-EAM total labelling of the Cn∘K2,m∪K1,n∪n−1/2K1.(ii)
For2≤n≡0mod2, obtain any S-a,0-EAM total labelling of Cn∘K2,m∪nP2.(iii)
For2≤n≡0mod2, obtain any S-a,0-EAM total labelling of Cn∘K2,m∪Pn+1.(iv)
For2≤n≡0mod2, determine S-a,0-EAM total labelling of G1, G2, G3, G4, G5, and G6 for any other magic constants (i.e., for any other value of a) than computed here.(v)
Forl,m and n positive integers, determine any S-a,0-EAM total labelling for the following networks:(1)
Cn∘K2,m∪K1,ℓ(2)
Cn∘K2,m∪lP2(3)
Cn∘K2,m∪PlOpen problems related to Theorems11 and 12 are given as follows:(i)
For2≤n≡0mod2, find any S-a,0-EAM total labelling of Cn∘H1.(ii)
For2≤n≡0mod2, find any S-a,0-EAM total labelling of Cn∘H2.(iii)
For3≤n≡1mod2, determine some S-a,0-EAM total labelling of Cn∘H1 and Cn∘H2 with a different magic constant than obtained here, i.e., for any other value of a.


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

This section consists of the synopsis (Section4.1) and 3-D graphical plots and comparison of the magic constants (a) and minimum edge weights (a′) of our findings (Section 4.2).


### 4.1. Synopsis

Table2 exhibits the computational results of our findings. The possible parameters for which we have determined S-a,0 and S-a′,2-EAM total labellings are indicated through parameters column head.Table 2
Synopsis of main theorems.
NetworkParametersa (d = 0)a′d=2PlanarV nonplanarCn∘K2,m∪K1,n∪n−1/2K1m≥2, odd n≥38n+3mn+26n+mn+3PlanarCn∘K2,m∪nP2m≥2, odd n≥31/217n+6mn+31/213n+2mn+5PlanarCn∘K2,m∪Pn+1m≥2, odd n≥31/213n+6mn+71/29n+2mn+9PlanarCn∘H1 and Cn∘H2n is odd1/245n+31/217n+5PlanarΓn∀n∈ℕ18n6n+4PlanarLn1∀n∈ℕ18n6n+4NonplanarLn2 and Ln3∀n∈ℕ30n−1210nNonplanar


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

Figure16 shows the graphical comparison of the magic constants (⟶A) and minimum edge weights (⟶B), in 3-D, of the networks M1≅Cn∘K2,m∪K1,n∪n−1/2K1,M2≅Cn∘K2,m∪nP2,M3≅Cn∘K2,m∪Pn+1, Cn∘H1, Cn∘H2, Γn, Ln1, Ln2, and Ln3, 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, Cn∘K2,m∪nP2 attains highest values with the increase of the values of the parameters.Figure 16
Comparison of the magic constants (⟶A) and the minimum edge weights (⟶B) of M1, M2, M3, Cn∘H1, Cn∘H2, Γn, Ln1, Ln2, and Ln3, for m≥2 and 3≤n≡1mod2.
(a)(b)Figure17 illustrates the relative 3-D comparison of the magic constants and minimum edge weights (corresponding to d=2) of the networks Cn∘K2,m∪K1,n∪n−1/2K1, Cn∘K2,m∪nP2, Cn∘K2,m∪Pn+1, Cn∘H1, Cn∘H2, Γn, Ln1, Ln2 and Ln3, for different values of the parameters.Figure 17
Simultaneous relative comparison of the magic constants and the minimum edge weights ofM1 (⟶A), M2 (⟶B), M3 (⟶C), {Cn∘H1, Cn∘H2} (⟶D), Γn (⟶E), {Ln1, Ln2, Ln3} (⟶F), for m≥2 & 3≤n≡1mod2
(a)(b)(c)(d)(e)(f)


## 4.1. Synopsis

Table2 exhibits the computational results of our findings. The possible parameters for which we have determined S-a,0 and S-a′,2-EAM total labellings are indicated through parameters column head.Table 2
Synopsis of main theorems.
NetworkParametersa (d = 0)a′d=2PlanarV nonplanarCn∘K2,m∪K1,n∪n−1/2K1m≥2, odd n≥38n+3mn+26n+mn+3PlanarCn∘K2,m∪nP2m≥2, odd n≥31/217n+6mn+31/213n+2mn+5PlanarCn∘K2,m∪Pn+1m≥2, odd n≥31/213n+6mn+71/29n+2mn+9PlanarCn∘H1 and Cn∘H2n is odd1/245n+31/217n+5PlanarΓn∀n∈ℕ18n6n+4PlanarLn1∀n∈ℕ18n6n+4NonplanarLn2 and Ln3∀n∈ℕ30n−1210nNonplanar


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

Figure16 shows the graphical comparison of the magic constants (⟶A) and minimum edge weights (⟶B), in 3-D, of the networks M1≅Cn∘K2,m∪K1,n∪n−1/2K1,M2≅Cn∘K2,m∪nP2,M3≅Cn∘K2,m∪Pn+1, Cn∘H1, Cn∘H2, Γn, Ln1, Ln2, and Ln3, 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, Cn∘K2,m∪nP2 attains highest values with the increase of the values of the parameters.Figure 16
Comparison of the magic constants (⟶A) and the minimum edge weights (⟶B) of M1, M2, M3, Cn∘H1, Cn∘H2, Γn, Ln1, Ln2, and Ln3, for m≥2 and 3≤n≡1mod2.
(a)(b)Figure17 illustrates the relative 3-D comparison of the magic constants and minimum edge weights (corresponding to d=2) of the networks Cn∘K2,m∪K1,n∪n−1/2K1, Cn∘K2,m∪nP2, Cn∘K2,m∪Pn+1, Cn∘H1, Cn∘H2, Γn, Ln1, Ln2 and Ln3, for different values of the parameters.Figure 17
Simultaneous relative comparison of the magic constants and the minimum edge weights ofM1 (⟶A), M2 (⟶B), M3 (⟶C), {Cn∘H1, Cn∘H2} (⟶D), Γn (⟶E), {Ln1, Ln2, Ln3} (⟶F), for m≥2 & 3≤n≡1mod2
(a)(b)(c)(d)(e)(f)


## 5. Conclusion

In the present article,(i)
We have designed S-a,0-EAM total labelling of the rooted product of cycle Cn and complete bipartite network K2,m taking its disjoint union with paths and stars. The findings are related to the open problem on nK2,m provided by Ngurah et al. in [48].(ii)
We have provided S-a,0-EAM total labelling of rooted product of Cn and pancyclic networks H1 and H2.(iii)
We have extended the result provided in [29] by Baig et al. through exhibiting S-a,0-EAM total labelling of pancyclic network Γn involving chains of C6.(iv)
We have exhibited S-a,0-EAM total labelling of symmetrically designed lattice networks Ln1, Ln2, and Ln3.(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)
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.



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