# Enumeration of the Edge Weights of Symmetrically Designed Graphs

Authors: Muhammad Javaid; Hafiz Usman Afzal; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2021)

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## Abstract

The idea of supera,0-edge-antimagic labeling of graphs had been introduced by Enomoto et al. in the late nineties. This article addresses super a,0-edge-antimagic labeling of a biparametric family of pancyclic graphs. We also present the aforesaid labeling on the disjoint union of graphs comprising upon copies of C4 and different trees. Several problems shall also be addressed in this article.


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## Body

## 1. Introduction

A graphΓV,E is a combination of two different sets, one is the set of vertices VΓ and the other is the set of connections between these vertices, termed as set of edges EΓ. A graph Γ can either be connected or comprises upon connected parts known as graphs’ components. The nonempty and simple graphs shall be considered here only all the way, consisting of VΓ, the set of vertices, and EΓ, the set of edges, having VΓ=p and EΓ=q. In this case, the graph Γ is called a p,q-graph. Additionally, [1] can be cited for the comprehension of the graph theoretic terminologies.A labeling is a function from the set of integers onto the components of a graph under certain constraints. The labeling is said to be total if it covers all components of the graph. If the labeling coversVΓ or EΓ only in the domain, then it is termed to be the vertex or edge labeling, respectively. The two important categories of labeling are magic and antimagic. The equal or unequal edge/vertex weights point towards, respectively, the magic and antimagic types of labeling.Throughout the article, the abbreviations given in Table1 are used.Table 1
Abbreviations being used onwards.
TerminologyAbbreviationa,d-edge-antimagic totala,d−EAMTSupera,d-edge-antimagic totalS−a,d−EAMTDefinition 1.
On aσ,ς-graph Γ=VΓ,EΓ, a bijection γ from VΓ∪EΓonto⟶1,2,…,σ+ς is the notion of a,d−EAMT labeling if we keep the restriction upon the edge weights γu+γuv+γv, ∀uv∈EG, which generates a consecutive integer sequence, with a being the minimum edge weight and d being the common difference. Γ is notioned as an a,d−EAMT graph, with the existence of such labeling.Definition 2.
If the smallest labels1,2,…,σ are allocated to the points (vertices) of the σ,ς-graph Γ in an a,d−EAMT labeling, then this labeling is addressed as S−a,d−EAMT labeling. And, Γ, in his scenario, is referred to be an S−a,d−EAMT graph.
The edge weight (minimum)a (Definitions 1 and 2) becomes a constant at d=0, ∀uv∈EΓ, and is called magic constant or magic sum of Γ.Definition 3.
A pancyclic graphΓV,E is a graph that contains the cycles of all orders up to VΓ.
The notion of magic labeling was highlighted by Sedlacek in the early sixties [2]. Later, Ringel and Hartsfield capitulated the idea of antimagic labeling with respect to vertex-sums of graphs in [3]. The idea of magic valuations of graphs had been brought by Kotzig and Rosa [4] which was indeed the graphs’ a,0−EAMT labeling (introduced by Ringel et. al. in the nineties [5]). Enomoto and Llado introduced the idea of S−a,0−EAMT labeling of graphs using the term super edge-magic labeling in [6]. In the early 21st century, Bertault and Simanjuntak brought out the graphs’ a,d−EAMT labeling [7]. The following notable and handy conjectures are included in the vicinity of graphs’ a,0−EAMT labeling.Conjecture 1 (see [4]).
Every tree admits ana,0−EAMTlabeling.Conjecture 2 (see [6]).
Every tree admits anS−a,0−EAMTlabeling.In the support of Conjecture2, certain classes of trees have been sorted out by scientists. For trees having maximum of seventeen vertices, in [8], this conjecture has been verified. For instance, the S−a,0−EAMT labeling on a class of trees termed as w-trees can be observed in [9]. Similarly, S−a,0−EAMT labeling on various classes consisting of subdivisions of trees can be seen in [10, 11]. Some derivations on vertex-antimagicness of regular graphs have been discussed in [12]. In [13], the same labeling for the union of unicyclic graphs and isolated vertices has been provided. Enomoto et al. proved [6] that if a p,q-graph Γ (simple) is S−a,0−EAMT, it implies q≤2p−3. In addition, they derived that Km,n is S−a,0−EAMT only if either m or n is equal to 1. It is derived in [14] that the combination of graphs in the form of union of K1,θ and K1,η is S−a,0−EAMT if θ=χη+1 or η=χθ+1. For only odd values of n, Cn is concreted to be S−a,0−EAMT [6]. The cycle of order 3 and cycle of order n≥6 are proven to be S−a,0−EAMT for even values of n. The generalized prism Dξ is proven to be S−a,0−EAMT for all odd values of ξ in [15]. In [16, 17], S−a,0−EAMT labeling of maximum symmetric generalization of prism and special networks with magic constant 3p has been exhibited, respectively. An extremely important result on S−a,0−EAMT graphs is as follows.Lemma 1 (see [15]).
Ap,q-graphGisS−a,0−EAMTif and only if there exists a bijective functionλ:VG⟶1,2,…,psuch that the setS=λu+λu|uv∈EGconsists ofqconsecutive integers. In such a case,λextends to anS−a,0−EAMTlabeling ofGwith magic constantt=p+q+s, wheres=minSandS=t−p+1,t−p+2,…,t−p+q.In Lemma1, the sum ζu+ζv is defined as edge sum for each edge uv∈EΓ. This lemma shall be used frequently in our findings, while it keeps this enough to allot the labels to merely the vertices of the network to capacitate the graph to be S−a,0−EAMT, where the edge-sums (consecutive) belong to ℕ. The given result is extremely relevant with regard to S−a,0−EAMT.Lemma 2 (see [18]).
A simple graphΓpossesses anS−a,0−EAMTlabeling⇔Γpossesses anS−a−EG+1,2−EAMTlabeling.


## 2. Main Results

In this section, we shall address our main findings. In Section2.1, we define an S−a,0−EAMT labeling on a pancyclic family of graphs, namely, Usmanian pancyclic graph UPnt. In Section 2.2, we design S−a,0−EAMT labeling on various disjoint unions of graphs comprising upon copies of C4 and various trees/forests. Throughout, i∈a,b represents a≤i≤b, for i,a,b∈ℕ, whereas ℕodd and ℕeven are the representations of odd and even natural numbers, respectively.


### 2.1. Usmanian Pancyclic GraphUPnt

In computer science, there is a similar importance of the networks having no cycles and networks having a range of cycles. The importance is similar for a network containing cycles of all lengths from one to the number of systems connected within. In this situation, the role of programmers becomes prominent to avoid hackers halting of data as there is a closed path between any two arbitrary computers corresponding to such networks. The first kind of network is termed as a tree (connected and acyclic) and later is known as pancyclic network (connected and containing every order’s cycle). This section deals with a family of pancyclic graphs denoted byUPnt, which is biparametric, and reveals that such complex structures are S−a,0−EAMT. We shall first introduce this structure as Definition 4.Definition 4.
We are defining the Usmanian pancyclic graph, denoted byUPnt, being a graph with VUPnt=tnandEUPnt=2tn−3, having the construction as follows (n≥3 being the order of the cycle Cnandt≥2 being the number of cycles):(1)
For evenn:Forn≡0mod4:(i)
Forn=4,(1)VUP4t=yi,zi:i∈1,t∪xi:i∈1,2t,EUP4t=zizi+1,yiyi+1:i∈1,t−1∪yix2i,zix2i:i∈1,t∪zix2i−1,yix2i−1:i∈1,t∪xixi+1:i∈1,2t−1.(ii)
Forn=8,(2)VUP8t=xi,vi:i∈1,t∪wi,yi,zi:i∈1,2tEUP8t=wiwi+1,yiyi+1,zizi+1:i∈2,2t−2,i∈ℕeven∪ziwi,yizi:i∈1,2t∪xiy2i−1,xiy2i,viw2i−1,viw2i:i∈1,t∪yiwi:i∈1,2t∪xivi:i∈1,t∪ziwi+1,yizi+1:i∈1,2t−1,i∈ℕodd.(iii)
Forn=12,(3)VUP12t=xij:i∈1,2t,2≤j≤6∪xij:i∈1,t,j=1,7,EUP12t=xijxi+1j:i∈2,2t−1,i∈ℕeven,3≤j≤5∪xijxij+1:i∈1,2t,2≤j≤5∪xi1x2i−12,xi1x2i2,xi7x2i−16,xi7x2i6:i∈1,t∪xi1xi7:i∈1,t∪xijxi8−j:i∈1,2t,2≤j≤6∪xi4xi7:i∈1,2t−1,i∈ℕodd∪xi2xi+15,xi3xi+16:i∈1,2t−1,i∈ℕodd∪xi1x2i4:i∈1,t.(iv)
Forn≥16,(4)VUPnt=xij:i∈1,2t,2≤j≤n2∪xij:i∈1,t,j=1,n+22,EUPnt=xijxi+1j:i∈2,2t−1,i∈ℕeven,n4≤j≤n+84∪xijxij+1:i∈1,2t,2≤j≤n−22∪xi1x2i−12,xi1x2i2,xin+2/2x2i−1n/2,xin+2/2x2in/2:i∈1,t∪xi1xin+2/2:i∈1,t∪xijxin/2+2−j:i∈1,2t,2≤j≤n2∪xijxin/2+5−j:i∈1,2t−1,i∈ℕodd,4≤j≤n+44∪xi2xi+1n−2/2,xi3xi+1n/2:i∈1,2t−1,i∈ℕodd∪xi1x2in−4/2:i∈1,t∪xijxin−2/2−j:i∈2,2t,i∈ℕeven,2≤j≤n−84.Forn≡2mod4:(i)
Forn=6,(5)VUP6t=xi,yi,zi:i∈1,2t,EUP6t=zizi+1,xixi+1:i∈1,2t−1∪yizi,xiyi:i∈1,2t∪yiyi+1:i∈1,2t−1∪xizi:i∈1,2t.(ii)
Forn=10,(6)VUP10t=xij:i∈1,2t,j∈1,5EUP10t=xijxij+1:i∈1,2t,1≤j≤4∪xi1xi+11,xi5xi+15:i∈1,2t−1,i∈ℕodd∪xijxi+1j:i∈2,2t−1,i∈ℕeven,2≤j≤4∪xi1xi5,xi2xi4:i∈1,2t∪xijxi+12+j:i∈1,2t−1,i∈ℕodd,1≤j≤3.(iii)
Forn≥14,(7)VUPnt=xij:i∈1,2t,1≤j≤n2,EUPnt=xijxij+1:i∈1,2t,1≤j≤n−22∪xi1xi+11,xin/2xi+1n/2:i∈1,2t−1,i∈ℕodd∪xijxi+1j:i∈2,2t−1,i∈ℕeven,n−24≤j≤n+64∪xijxin+2/2−j:i∈1,2t,1≤j≤n−24∪xijxi+1n−6/2+j:i∈1,2t−1,i∈ℕodd,1≤j≤3∪xijxin+8/2−j:i∈1,2t−1,i∈ℕodd,4≤j≤n+24∪xijxin−4/2−j:i∈2,2t,i∈ℕeven,1≤j≤n−104.(2)
For oddn:Forn≡1mod4:(i)
Forn=5,(8)VUP5t=zi,yi:i∈1,2t∪xi:i∈1,t,EUP5t=zizi+1:i∈1,2t−1,i∈ℕodd∪yizi:i∈1,2t∪xiy2i−1,xiy2i:i∈1,t∪yiyi+1:i∈2,2t−2,i∈ℕeven∪xiz2i+1:i∈1,t−1∪xiz2i−2:i∈2,t∪xiz2i−1,xiz2i:i∈1,t.(ii)
Forn=9,(9)VUP9t=wi,vi,yi,zi:i∈1,2t∪xi:i∈1,t,EUP9t=vivi+1:i∈1,2t−1,i∈ℕodd∪yizi,ziwi,wivi:i∈1,2t∪xiy2i−1,xiy2i:i∈1,t∪yiwi+1,zizi+1,wiyi+1:i∈2,2t−2,i∈ℕeven∪yiwi,yivi:i∈1,2t∪xiv2i−1,xiv2i:i∈1,t.(iii)
Forn≥13,(10)VUPnt=xij:i∈1,2t,j∈2,n+12∪xi1:i∈1,t,EUPnt=xijxij+1:i∈1,2t,2≤j≤n−12∪xi1x2i−12,xi1x2i2:i∈1,t∪xin+1/2xi+1n+1/2:1≤i≤2t−1,i∈ℕodd∪xin−1/4xi+1n+7/4,xin+7/4xi+1n−1/4,xin+3/4xi+1n+3/4:2≤i≤2t−1,i∈ℕeven∪xijxin+3/2−j:i∈1,2t,2≤j≤n−14∪xijxin+5/2−j:i∈1,2t,2≤j≤n−14∪xi1x2i−1n+1/2,xi1x2in+1/2:i∈1,t.Forn≡3mod4:(i)
Forn=3:DefineUP3t≅Pt×C3 as follows:(11)VUP3t=zi,xi,yi:i∈1,t,EUP3t=zizi+1,xixi+1,yiyi+1:i∈1,t−1∪xizi,xiyi,yizi:i∈1,t.(ii)
Forn=7,(12)VUP7t=zi,wi,yi:i∈1,2t∪xi:i∈1,t,EUP7t=wiwi+1:i∈1,2t−1,i∈ℕodd∪ziwi,yizi:i∈1,2t∪xiy2i−1,xiy2i:i∈1,t∪zizi+1,yiwi+1,wiyi+1:i∈2,2t−2,i∈ℕeven∪yiwi:i∈1,2t∪xiw2i−1,xiw2i:i∈1,2t.(iii)
Forn≥11,(13)VUPnt=xij:i∈1,2t,2≤j≤n+12∪xi1:i∈1,t,EUPnt=xijxij+1:i∈1,2t,2≤j≤n−12∪xi1x2i−12,xi1x2i2:i∈1,t∪xin+1/2xi+1n+1/2:i∈1,2t−1,i∈ℕodd∪xin+1/4xi+1n+9/4,xin+5/4xi+1n+5/4,xin+9/4xi+1n+1/4:i∈2,2t−1,i∈ℕeven∪xijxin+5/2−j:i∈1,2t,2≤j≤n+14∪xijxin+3/2−j:i∈1,2t,2≤j≤n−34∪xi1x2i−1n+1/2,xi1x2in+1/2:i∈1,t.Theorem 1.
The pancyclic graphUPnt is S−a,0−EAMT admitting a=3tn, ∀t and n≡0mod2.Proof.
We discuss here two cases as per Definition4.Forn≡0mod4:(i)
Forn=4:We define a labelingψ1′:VUP4t⟶1,2,…,4t as follows:(14)ψ1′xi=2i−1:i∈1,2t−1,i∈ℕodd;2i:2≤i≤2t,i∈ℕeven;ψ1′yi=4i−2:i∈1,t,ψ1′zi=4i−1:i∈1,t.The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers3,4,…,8t−1. As per Lemma 1, ψ1′ is extendable to an S−a,0−EAMT labeling of UP4t with magic constant a=12t.(ii)
Forn=8:We define a labelingψ1″:VUP8t⟶1,2,…,8t as follows:(15)ψ1″xi=8i−3:1≤i≤t,ψ1″yi=4i−1:1≤i≤2t,ψ1″zi=4i−3:i∈1,2t−1,i∈ℕodd;4i:2≤i≤2t,i∈ℕeven;ψ1″wi=4i−2:i∈1,2t,ψ1″vi=8i−4:i∈1,t.The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers3,4,…,16t−1. In the light of Lemma 1, ψ1′′ constitutes an S−a,0−EAMT labeling of UP8t admitting a=24t.(iii)
Forn≥12:We define a labelingψ1:VUPnt⟶1,2,…,tn as follows:(16)ψ1xij=2ni−n+22:i∈1,tandj=1;ni−4j+62:i∈1,2t−1,j∈2,n+44,i∈ℕodd;ni+4j−n−22:i∈2,2t,j∈2,n4,i∈ℕeven;ni+4j−n−42:i∈2,2t,j=n+44,i∈ℕeven;ni+4j−2n−42:i∈1,2t−1,j∈n+84,n2,i∈ℕodd;ni−4j+n+42:i∈2,2t,j∈n+84,n2,i∈ℕeven;2ni+4j−3n−42:i∈1,tandj=n+22.
With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence setΔ=3,4,…,2tn−1. ψ1 is extendible to S−a,0−EAMT labeling of UPnt,n≥12, according to Lemma 1, with a=3tn.Forn≡2mod4:(i)
Forn=6:The labelingψ2′:VUP6t⟶1,2,…,6t is being defined as follows:(17)ψ2′xi=3i:1≤i≤2t−1,i∈ℕodd;3i−1:2≤i≤2t,i∈ℕeven;ψ2′yi=3i−2:1≤i≤2t−1,i∈ℕodd;3i:2≤i≤2t,i∈ℕeven;ψ2′zi=3i−1:1≤i≤2t−1,i∈ℕodd;3i−2:2≤i≤2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence3,4,…,12t−1. As per Lemma 1, ψ2′ is extendable towards S−a,0−EAMT labeling of UP6t with a=18t.(ii)
Forn≥10:Define a labelingψ2:VUPnt⟶1,2,…,tn as the following function:(18)ψ2xij=ni−4j+42:i∈1,2t−1,j∈1,n+24,i∈ℕodd;ni+4j−n2:i∈2,2t,j∈1,n−24,i∈ℕeven;ni+4j−2n−22:i∈1,2t−1,j∈n+64,n2,i∈ℕodd;ni−4j+n+22:i∈2,2t,j∈n+24,n2,i∈ℕeven.
With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence set3,4,…,2tn−1. ψ2 constitutes S−a,0−EAMT labeling of UPnt,n≥10, according to Lemma 1, admitting a=3tn.Theorem 2.
The pancyclic graphUPnt is S−a,0−EAMT with magic sum a=3tn, for all t and n≡1mod2.Proof.
We discuss here two cases.Forn≡1mod4:(i)
Forn=5:Define a labelingψ3′:VUP5t⟶1,2,…,5t as follows:(19)ψ3′xi=5i−2:i∈1,t,ψ3′yi=125i−3:i∈1,2t−1,i∈ℕodd;125i:i∈2,2t,i∈ℕeven;ψ3′zi=125i−1:i∈1,2t−1,i∈ℕodd;125i−2:i∈2,2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence3,4,…,10t−1. ψ3′ is extendable to S−a,0−EAMT labeling of UP5t having a=15t (as per Lemma 1).(ii)
Forn=9:We construct a labelingψ3′:VUP9t⟶1,2,…,9t as follows:(20)ψ3″xi=9i−4:1≤i≤t,ψ3″yi=129i−3:i∈1,2t−1,i∈ℕodd;129i−4:i∈2,2t,i∈ℕeven;ψ3′zi=129i−7:i∈1,2t−1,i∈ℕodd;129i:i∈2,2t,i∈ℕeven;ψ3″wi=129i−5:i∈1,2t−1,i∈ℕodd;129i−2:i∈2,2t,i∈ℕeven;ψ3″vi=129i−1:i∈1,2t−1,i∈ℕodd;129i−6:i∈2,2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a natural numbers’ sequence3,4,…,18t−1. ψ3′′ is extendable to S−a,0−EAMT labeling of UP9t, by Lemma 1, with the admittance of a=27t.(iii)
Forn≥13:We are going to construct a labelingψ3:VUPnt⟶1,2,…,tn as follows:(21)ψ3xij=2ni−n+12:i∈1,t,j=1,ni−4j+52:i∈1,2t−1,j∈2,n+34,i∈ℕodd,ni+4j−n−32:i∈2,2t,j∈2,n+34,i∈ℕeven,ni+4j−2n−32:i∈1,2t−1,j∈n+74,n+12,i∈ℕodd,ni−4j+n+52:i∈2,2t,j∈n+74,n+12,i∈ℕeven.With the abovementioned scheme, the edge-sums being generated forms a consecutive integer sequence set3,4,…,2tn−1. Once again, ψ3 extends to a S−a,0−EAMT labeling of UPnt,n≥13 having a=3tn by Lemma 1.Forn≡3mod4:(i)
Forn=3:We define a labelingψ4′:VUP3t⟶1,2,…,3t as follows:(22)ψ4′xi=3i−2:1≤i≤t−1,i∈ℕodd,3i−1:2≤i≤t,i∈ℕeven,ψ4′yi=3i:1≤i≤t−1,i∈ℕodd,3i−2:2≤i≤t,i∈ℕeven,ψ4′zi=3i−1:1≤i≤t−1,i∈ℕodd,3i:2≤i≤t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers3,4,…,6t−1. Now, ψ4′ is extendable to S−a,0−EAMT labeling of UP3t≅Pt×C3 [15] with a=9t according to Lemma 1.(ii)
Forn=7:A labelingψ4″:VUP7t⟶1,2,…,7t is being defined as follows:(23)ψ4″xi=7i−3:1≤i≤t,ψ4″yi=127i−3:i∈1,2t−1,i∈ℕodd,127i−2:i∈2,2t,i∈ℕeven,ψ4″zi=127i−5:i∈1,2t−1,i∈ℕodd,127i:i∈2,2t,i∈ℕeven,ψ4″wi=127i−1:i∈1,2t−1,i∈ℕodd,127i−4:i∈2,2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence3,4,…,14t−1. Now, ψ4′′ is extendable to S−a,0−EAMT labeling of UP7t, according to Lemma 1 having a=21t.(iii)
Forn≥11:The labelingψ4:VUPnt⟶1,2,…,tn is constructed as follows:(24)ψ4xij=2ni−n+12:i∈1,t,j=1,ni−4j+52:i∈1,2t−1,j∈2,n+14,i∈ℕodd,ni+4j−n−32:i∈2,2t,j∈2,n+14,i∈ℕeven,ni+4j−2n−32:i∈1,2t−1,j∈n+54,n+12,i∈ℕodd,ni−4j+n+52:i∈2,2t,j∈n+54,n+12,i∈ℕeven.With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence set3,4,…,2tn−1. ψ4 is extendable to an S−a,0−EAMT labeling of UPnt,n≥11, under the light of Lemma 1, admitting a=3tn.
A direct outcome of Lemma2 is as follows.Theorem 3.
The pancyclic graphUPnt is S−tn+4,2−EAMT, for all t and n.


### 2.2.S−a,0−EAMT Labeling of Disjoint Union of C4 with Trees

It is a well-known fact that the graphC4 is not S−a,0−EAMT [6], and work is still in progress in order to determine if its disjoint copies are S−a,0−EAMT. In this section, we shall provide an S−a,0−EAMT labeling of disjoint copies of C4 with various trees in the form of several results. This will give a support to researchers to carry out their work to determine the aforesaid labeling of the disjoint copies of C4. Throughout this section, the union will represent a disjoint union of graphs only.Theorem 4.
For oddm, the graph mC4∪2K1,m∪7m−3/2K1 acquires an S−a,0−EAMT labeling admitting a=21m+2.Proof.
Consider a graphmC4∪2K1,m∪7m−3/2K1 with vertex and edge sets:(25)VΛ1=x1i,x2i:i∈1,m∪yi,zi:i∈1,m∪ki:1≤i≤2m∪li:1≤i≤7m−32∪c1,c2,EΛ1=yix1i,yix2i:1≤i≤m∪zix1i,zix2i:1≤i≤m∪c1ki:1≤i≤m∪c2ki:m+1≤i≤2m.
Ifp=VΛ1=19m+1/2 and q=EΛ1=6m, we sketch a labeling f1:VΛ1⟶1,2,…,19m+1/2 as follows:(26)f1x1i=127m+i,1≤i≤m;i≡1,mod2,126m+i,2≤i≤m−1;i≡0,mod2,f1x2i=5m−i−1:1≤i≤m,f1yi=i+4m+12,1≤i≤m;i≡1,mod2,i+5m+12,2≤i≤m−1;i≡0,mod2,f1zi=12i+10m+1,i∈1,m;i∈ℕodd,12i+11m+1,i∈2,m−1;i∈ℕeven,f1ki=i:i∈1,2m,f1c1=15m+12,f1c2=19m+12,f1li=6m+i,1≤i≤3m−12,i+6m+1,i∈3m+12,7m−32.
The edge-sums’ sets constituted by the abovementioned design generates a consecutive sequence of integer11m+3/2,11m+5/2,…,23m+1/2. Under the shadow of Lemma 1, f1 accredits an S−a,0−EAMT labeling of Λ1 having a=21m+2.Theorem 5.
For oddm, the graph mC4∪2mK2∪2mK1 acquires an S−a,0−EAMT labeling having a=43m+3/2.Proof.
Consider the graphΛ2≅mC4∪2mK2∪2mK1, for odd m, with the following vertex-edge connections:(27)VΛ2=x1i,x2i:i∈1,m∪zi,yi:1≤i≤m∪qi,pi:i∈1,2m∪li:i∈1,2m,EΛ2=yix1i,yix2i:1≤i≤m∪zix1i,zix2i:1≤i≤m∪qipi:i∈1,2m.
Here, order isp=10m and size is q=6m. Now, we design a labeling f2:VΛ2⟶1,2,…,10m as follows:(28)f2x1i=127m+i,i∈1,m;i∈ℕodd,126m+i,i≤2,m−1;i∈ℕeven,f2x2i=5m−i−1:i∈1,m,f2yi=12i+4m+1,i∈1,…,m;i∈ℕodd,12i+5m+1,i∈2,…,m−1;i∈ℕeven,f2zi=12i+10m+1,i∈1,m;i∈ℕodd,12i+11m+1,i∈2,m−1;i∈ℕeven,f2pi=i:i∈1,2m,f2qi=1215m−i+2:i∈1,m;i∈ℕodd,1216m−i+2:i∈2,m−1;i∈ℕeven,1221m−i+2:i∈m+2,2m−1;i∈ℕodd,1220m−i+2:i∈m+1,2m;i∈ℕeven,f2li=i+6m:i∈1,m,i+7m:i∈m+1,2m.
The edge-sums’ set constituted by the schemef2 generates a sequence consisting of consecutive integer 11m+3/2,11m+5/2,…,23m+1/2. Under the shadow of Lemma 1, f2 constitutes an S−a,0−EAMT labeling of Λ2 with magic sum a=43m+3/2.Theorem 6.
For oddm, the graph mC4∪2Pm+1∪5m−1/2K1 acquires an S−a,0−EAMT labeling having a=18m+5.Proof.
LetΛ3≅mC4∪2Pm+1∪5m−1/2K1, where m is odd, having vertex and edge sets interlinked:(29)VΛ3=x1i,x2i:i∈1,m∪zi,yi:i∈1,m∪pi:i∈1,m+1∪qi:i∈1,m+1∪li:i∈1,5m−12,EΛ3=yix1i,yix2i:i∈1,m∪zix1i,zix2i:i∈1,m∪pipi+1:i∈1,m∪qiqi+1:i∈1,m.
We havep=17m+3/2 and q=6m. A labeling f3:VΛ3⟶1,2,…,17m+3/2 is being defined as follows:(30)f3x1i=125m+i+2,i∈1,m;i∈ℕodd,124m+i+2,i∈2,m−1;i∈ℕeven,f3x2i=4m−i+2:1≤i≤m,f3yi=2m+i+32,i∈1,m;i∈ℕodd,3m+i+32,i∈2,m−1;i∈ℕeven,f3zi=8m+i+32,i∈1,…,m;i∈ℕodd,9m+i+32,i∈2,…,m−1;i∈ℕeven,f3pi=i+12,i∈1,m;i∈ℕodd,11m+i+32,i∈2,m+1;i∈ℕeven,f3qi=12i+m+2,i∈1,m;i∈ℕodd,1216m+i+2,i∈2,m+1;i∈ℕeven,f3li=i+5m+1:1≤i≤m+12,2i+11m+32:m+32≤i≤5m−12.
The edge-sums’ set constituted by the schemef3 generates a sequence consisting of consecutive integer 7m+7/2,7m+9/2,…,19m+5/2. Under the shadow of Lemma 1, f3 constitutes an S−a,0−EAMT of the graph Λ3 with magic constant a=18m+5.Theorem 7.
For oddm, mC4∪2m−2K2∪P4∪2mK1 possesses S−a,0−EAMT labeling having a=43m+5/2.Proof.
Withm taken odd, consider Λ4≅mC4∪2m−2K2∪P4∪2mK1, having vertex-edge connections:(31)VΛ4=x1i,x2i:i∈1,m∪yi,zi:i∈1,m∪li:1≤i≤2m∪qi,pi:i∈1,2m−1∪ti:i∈1,4,EΛ4=yix1i,yix2i:1≤i≤m∪zix1i,zix2i:i∈1,m∪piqi:i∈1,2m−1∪titi+1:1≤i≤3.
Here, we havep=10m and q=6m+1. A labeling function f:VΛ4⟶1,2,…,10m is being defined as follows:(32)f4x1i=127m+i:i∈1,m;i∈ℕodd,126m+i:i∈2,m−1;i∈ℕeven,f4x2i=5m−i−1:1≤i≤m,f4yi=i+4m+12:i∈1,…,m;i∈ℕodd,i+5m+12:i∈2,…,m−1;i∈ℕeven,f4zi=i+10m+12:i∈1,…,m;i∈ℕodd,i+11m+12:i∈2,…,m−1;i∈ℕeven,f4pi=i:i∈1,2m−1,f4qi=15m−i+22:i∈1,m;i∈ℕodd;16m−i+22:i∈2,m−1;i∈ℕeven;21m−i+22:i∈m+2,2m−3;i∈ℕodd;20m−i+22:iim+1,…,2m−1;i∈ℕeven;f4ti=2m−1:i=1;19m+32:i=2;2m:i=3;9m+1:i=4;f4li=i+6m:1≤i≤m;i+7m:i∈m+1,2m.
The edge-sums’ set constituted by the schemef4 generates a sequence consisting of consecutive integer 11m+3/2,11m+5/2,…,23m+3/2. Lemma 1 implies that f4 extends to an S−a,0−EAMT labeling of Λ4 with a=43m+5/2.Theorem 8.
For oddm, mC4∪Sm−1,m∪5m−1/2K1 possesses an S−a,0−EAMT labeling with a=29m+1.Proof.
Consider the graphΛ5≅mC4∪∪Sm−1,m∪5m−1/2K1 with vertex-edge connections as follows:(33)VΛ5=x1i,x2i:i∈1,m∪zi,yi:i∈1,m∪si,ti:i∈1,m∪li:i∈1,5m−12∪c,EΛ5=yix1i,yix2i:i∈1,m∪zix1i,zix2i:i∈1,m∪cti:i∈1,m∪tmsi:i∈1,m.
Then,p=17m+1/2 and q=6m. Again, a labeling f5:VΛ5⟶1,2,…,17m+1/2 is being defined as follows:(34)f5x1i=125m+i:i∈1,…,m;i∈ℕodd;124m+i:i∈2,m−1;i∈ℕeven,f5x2i=4m−i−1:i∈1,m,f5yi=i+2m+12:i∈1,…,m;i∈ℕodd;i+3m+12:i∈2,…,m−1;i∈ℕeven,f5zi=i+8m+12:i∈1,m;i∈ℕodd;i+9m+12:i∈2,m−1;i∈ℕeven,f5ti=i:1≤i≤m;f5si=2i+15m+12:1≤i≤m;fc=11m+12,f5li=i+5m:i∈1,m−12;i+5m+1:i∈m+12,5m−12.
The edge-sums’ set constituted by the schemef2 generates a sequence consisting of consecutive integer 7m+3/2,7m+5/2,…,19m+1/2. Under the shadow of Lemma 1, f5 constitutes to an S−a,0−EAMT labeling of Λ1 admitting a=29m+1.
The following results are direct consequences of Lemma2, from Theorems 4–8.Theorem 9.
For oddm, the graph mC4∪2K1,m∪7m−3/2K1 admits an S−15m+3,2−EAMT labeling.Theorem 10.
For oddm, the graph mC4∪2mP2∪2mK1 admits an S−31m+5/2,2−EAMT labeling.Theorem 11.
For oddm, the graph mC4∪2Pm+1∪5m−1/2K1 admits an S−12m+6,2−EAMT labeling.Theorem 12.
For oddm, mC4∪2m−2P2∪P4∪2mK1 admits an S−31m+5/2,2−EAMT labeling.Theorem 13.
For oddm, mC4∪Sm−1,m∪5m−1/2K1 admits an S−12m+3,2−EAMT labeling.


### 2.3. Examples and Proposed Open Problems

AnS−126,0−EAMT labeling of the graph UPnt is being presented in Figure 1(a), corresponding to the parameters t=3 and n=14. Furthermore, Figure 1(b) presents an S−96,0−EAMT labeling of UPnt corresponding to t=4 and n=8. Here, it can be observed that the value of the magic constant is perfect as per our depiction in Theorem 1.Figure 1
AnS−126,0−EAMT and S−96,0−EAMT labeling of the pancyclic graphs (a)UP143 and (b)UP84.
(a)(b)Figures2(a) and 2(b) illustrate Theorem 2 by providing S−117,0−EAMT and S−84,0−EAMT labeling of the graph UPnt.Figure 2
AnS−117,0−EAMT and S−84,0−EAMT labeling of the pancyclic graphs (a)UP133 and (b)UP74.
(a)(b)Figures3–7 are the illustrations of Theorems 4–8, respectively, for particular values of the parameters involved.Figure 3
AnS−107,0−EAMT labeling of the graph 5C4∪2K1,5∪16K1.Figure 4
AnS−109,0−EAMT labeling of the graph 5C4∪10K2∪10K1.Figure 5
AnS−95,0−EAMT labeling of the graph 5C4∪2P6∪12K1.Figure 6
AnS−109,0−EAMT labeling of the graph 5C4∪8K2∪P4∪10K1.Figure 7
AnS−92,0−EAMT labeling of the graph 5C4∪S4,5∪12K1.The open problems related to Section2.2 are as follows:(i)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2K1,m∪7m−3/2K1(ii)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2mK2∪2mK1(iii)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2Pm+1∪5m−1/2K1(iv)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2m−2K2∪P4∪2mK1(v)
For evenm, determine any S−a,0−EAMT labeling of mC4∪Sm−1,m∪5m−1/2K1


## 2.1. Usmanian Pancyclic GraphUPnt

In computer science, there is a similar importance of the networks having no cycles and networks having a range of cycles. The importance is similar for a network containing cycles of all lengths from one to the number of systems connected within. In this situation, the role of programmers becomes prominent to avoid hackers halting of data as there is a closed path between any two arbitrary computers corresponding to such networks. The first kind of network is termed as a tree (connected and acyclic) and later is known as pancyclic network (connected and containing every order’s cycle). This section deals with a family of pancyclic graphs denoted byUPnt, which is biparametric, and reveals that such complex structures are S−a,0−EAMT. We shall first introduce this structure as Definition 4.Definition 4.
We are defining the Usmanian pancyclic graph, denoted byUPnt, being a graph with VUPnt=tnandEUPnt=2tn−3, having the construction as follows (n≥3 being the order of the cycle Cnandt≥2 being the number of cycles):(1)
For evenn:Forn≡0mod4:(i)
Forn=4,(1)VUP4t=yi,zi:i∈1,t∪xi:i∈1,2t,EUP4t=zizi+1,yiyi+1:i∈1,t−1∪yix2i,zix2i:i∈1,t∪zix2i−1,yix2i−1:i∈1,t∪xixi+1:i∈1,2t−1.(ii)
Forn=8,(2)VUP8t=xi,vi:i∈1,t∪wi,yi,zi:i∈1,2tEUP8t=wiwi+1,yiyi+1,zizi+1:i∈2,2t−2,i∈ℕeven∪ziwi,yizi:i∈1,2t∪xiy2i−1,xiy2i,viw2i−1,viw2i:i∈1,t∪yiwi:i∈1,2t∪xivi:i∈1,t∪ziwi+1,yizi+1:i∈1,2t−1,i∈ℕodd.(iii)
Forn=12,(3)VUP12t=xij:i∈1,2t,2≤j≤6∪xij:i∈1,t,j=1,7,EUP12t=xijxi+1j:i∈2,2t−1,i∈ℕeven,3≤j≤5∪xijxij+1:i∈1,2t,2≤j≤5∪xi1x2i−12,xi1x2i2,xi7x2i−16,xi7x2i6:i∈1,t∪xi1xi7:i∈1,t∪xijxi8−j:i∈1,2t,2≤j≤6∪xi4xi7:i∈1,2t−1,i∈ℕodd∪xi2xi+15,xi3xi+16:i∈1,2t−1,i∈ℕodd∪xi1x2i4:i∈1,t.(iv)
Forn≥16,(4)VUPnt=xij:i∈1,2t,2≤j≤n2∪xij:i∈1,t,j=1,n+22,EUPnt=xijxi+1j:i∈2,2t−1,i∈ℕeven,n4≤j≤n+84∪xijxij+1:i∈1,2t,2≤j≤n−22∪xi1x2i−12,xi1x2i2,xin+2/2x2i−1n/2,xin+2/2x2in/2:i∈1,t∪xi1xin+2/2:i∈1,t∪xijxin/2+2−j:i∈1,2t,2≤j≤n2∪xijxin/2+5−j:i∈1,2t−1,i∈ℕodd,4≤j≤n+44∪xi2xi+1n−2/2,xi3xi+1n/2:i∈1,2t−1,i∈ℕodd∪xi1x2in−4/2:i∈1,t∪xijxin−2/2−j:i∈2,2t,i∈ℕeven,2≤j≤n−84.Forn≡2mod4:(i)
Forn=6,(5)VUP6t=xi,yi,zi:i∈1,2t,EUP6t=zizi+1,xixi+1:i∈1,2t−1∪yizi,xiyi:i∈1,2t∪yiyi+1:i∈1,2t−1∪xizi:i∈1,2t.(ii)
Forn=10,(6)VUP10t=xij:i∈1,2t,j∈1,5EUP10t=xijxij+1:i∈1,2t,1≤j≤4∪xi1xi+11,xi5xi+15:i∈1,2t−1,i∈ℕodd∪xijxi+1j:i∈2,2t−1,i∈ℕeven,2≤j≤4∪xi1xi5,xi2xi4:i∈1,2t∪xijxi+12+j:i∈1,2t−1,i∈ℕodd,1≤j≤3.(iii)
Forn≥14,(7)VUPnt=xij:i∈1,2t,1≤j≤n2,EUPnt=xijxij+1:i∈1,2t,1≤j≤n−22∪xi1xi+11,xin/2xi+1n/2:i∈1,2t−1,i∈ℕodd∪xijxi+1j:i∈2,2t−1,i∈ℕeven,n−24≤j≤n+64∪xijxin+2/2−j:i∈1,2t,1≤j≤n−24∪xijxi+1n−6/2+j:i∈1,2t−1,i∈ℕodd,1≤j≤3∪xijxin+8/2−j:i∈1,2t−1,i∈ℕodd,4≤j≤n+24∪xijxin−4/2−j:i∈2,2t,i∈ℕeven,1≤j≤n−104.(2)
For oddn:Forn≡1mod4:(i)
Forn=5,(8)VUP5t=zi,yi:i∈1,2t∪xi:i∈1,t,EUP5t=zizi+1:i∈1,2t−1,i∈ℕodd∪yizi:i∈1,2t∪xiy2i−1,xiy2i:i∈1,t∪yiyi+1:i∈2,2t−2,i∈ℕeven∪xiz2i+1:i∈1,t−1∪xiz2i−2:i∈2,t∪xiz2i−1,xiz2i:i∈1,t.(ii)
Forn=9,(9)VUP9t=wi,vi,yi,zi:i∈1,2t∪xi:i∈1,t,EUP9t=vivi+1:i∈1,2t−1,i∈ℕodd∪yizi,ziwi,wivi:i∈1,2t∪xiy2i−1,xiy2i:i∈1,t∪yiwi+1,zizi+1,wiyi+1:i∈2,2t−2,i∈ℕeven∪yiwi,yivi:i∈1,2t∪xiv2i−1,xiv2i:i∈1,t.(iii)
Forn≥13,(10)VUPnt=xij:i∈1,2t,j∈2,n+12∪xi1:i∈1,t,EUPnt=xijxij+1:i∈1,2t,2≤j≤n−12∪xi1x2i−12,xi1x2i2:i∈1,t∪xin+1/2xi+1n+1/2:1≤i≤2t−1,i∈ℕodd∪xin−1/4xi+1n+7/4,xin+7/4xi+1n−1/4,xin+3/4xi+1n+3/4:2≤i≤2t−1,i∈ℕeven∪xijxin+3/2−j:i∈1,2t,2≤j≤n−14∪xijxin+5/2−j:i∈1,2t,2≤j≤n−14∪xi1x2i−1n+1/2,xi1x2in+1/2:i∈1,t.Forn≡3mod4:(i)
Forn=3:DefineUP3t≅Pt×C3 as follows:(11)VUP3t=zi,xi,yi:i∈1,t,EUP3t=zizi+1,xixi+1,yiyi+1:i∈1,t−1∪xizi,xiyi,yizi:i∈1,t.(ii)
Forn=7,(12)VUP7t=zi,wi,yi:i∈1,2t∪xi:i∈1,t,EUP7t=wiwi+1:i∈1,2t−1,i∈ℕodd∪ziwi,yizi:i∈1,2t∪xiy2i−1,xiy2i:i∈1,t∪zizi+1,yiwi+1,wiyi+1:i∈2,2t−2,i∈ℕeven∪yiwi:i∈1,2t∪xiw2i−1,xiw2i:i∈1,2t.(iii)
Forn≥11,(13)VUPnt=xij:i∈1,2t,2≤j≤n+12∪xi1:i∈1,t,EUPnt=xijxij+1:i∈1,2t,2≤j≤n−12∪xi1x2i−12,xi1x2i2:i∈1,t∪xin+1/2xi+1n+1/2:i∈1,2t−1,i∈ℕodd∪xin+1/4xi+1n+9/4,xin+5/4xi+1n+5/4,xin+9/4xi+1n+1/4:i∈2,2t−1,i∈ℕeven∪xijxin+5/2−j:i∈1,2t,2≤j≤n+14∪xijxin+3/2−j:i∈1,2t,2≤j≤n−34∪xi1x2i−1n+1/2,xi1x2in+1/2:i∈1,t.Theorem 1.
The pancyclic graphUPnt is S−a,0−EAMT admitting a=3tn, ∀t and n≡0mod2.Proof.
We discuss here two cases as per Definition4.Forn≡0mod4:(i)
Forn=4:We define a labelingψ1′:VUP4t⟶1,2,…,4t as follows:(14)ψ1′xi=2i−1:i∈1,2t−1,i∈ℕodd;2i:2≤i≤2t,i∈ℕeven;ψ1′yi=4i−2:i∈1,t,ψ1′zi=4i−1:i∈1,t.The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers3,4,…,8t−1. As per Lemma 1, ψ1′ is extendable to an S−a,0−EAMT labeling of UP4t with magic constant a=12t.(ii)
Forn=8:We define a labelingψ1″:VUP8t⟶1,2,…,8t as follows:(15)ψ1″xi=8i−3:1≤i≤t,ψ1″yi=4i−1:1≤i≤2t,ψ1″zi=4i−3:i∈1,2t−1,i∈ℕodd;4i:2≤i≤2t,i∈ℕeven;ψ1″wi=4i−2:i∈1,2t,ψ1″vi=8i−4:i∈1,t.The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers3,4,…,16t−1. In the light of Lemma 1, ψ1′′ constitutes an S−a,0−EAMT labeling of UP8t admitting a=24t.(iii)
Forn≥12:We define a labelingψ1:VUPnt⟶1,2,…,tn as follows:(16)ψ1xij=2ni−n+22:i∈1,tandj=1;ni−4j+62:i∈1,2t−1,j∈2,n+44,i∈ℕodd;ni+4j−n−22:i∈2,2t,j∈2,n4,i∈ℕeven;ni+4j−n−42:i∈2,2t,j=n+44,i∈ℕeven;ni+4j−2n−42:i∈1,2t−1,j∈n+84,n2,i∈ℕodd;ni−4j+n+42:i∈2,2t,j∈n+84,n2,i∈ℕeven;2ni+4j−3n−42:i∈1,tandj=n+22.
With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence setΔ=3,4,…,2tn−1. ψ1 is extendible to S−a,0−EAMT labeling of UPnt,n≥12, according to Lemma 1, with a=3tn.Forn≡2mod4:(i)
Forn=6:The labelingψ2′:VUP6t⟶1,2,…,6t is being defined as follows:(17)ψ2′xi=3i:1≤i≤2t−1,i∈ℕodd;3i−1:2≤i≤2t,i∈ℕeven;ψ2′yi=3i−2:1≤i≤2t−1,i∈ℕodd;3i:2≤i≤2t,i∈ℕeven;ψ2′zi=3i−1:1≤i≤2t−1,i∈ℕodd;3i−2:2≤i≤2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence3,4,…,12t−1. As per Lemma 1, ψ2′ is extendable towards S−a,0−EAMT labeling of UP6t with a=18t.(ii)
Forn≥10:Define a labelingψ2:VUPnt⟶1,2,…,tn as the following function:(18)ψ2xij=ni−4j+42:i∈1,2t−1,j∈1,n+24,i∈ℕodd;ni+4j−n2:i∈2,2t,j∈1,n−24,i∈ℕeven;ni+4j−2n−22:i∈1,2t−1,j∈n+64,n2,i∈ℕodd;ni−4j+n+22:i∈2,2t,j∈n+24,n2,i∈ℕeven.
With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence set3,4,…,2tn−1. ψ2 constitutes S−a,0−EAMT labeling of UPnt,n≥10, according to Lemma 1, admitting a=3tn.Theorem 2.
The pancyclic graphUPnt is S−a,0−EAMT with magic sum a=3tn, for all t and n≡1mod2.Proof.
We discuss here two cases.Forn≡1mod4:(i)
Forn=5:Define a labelingψ3′:VUP5t⟶1,2,…,5t as follows:(19)ψ3′xi=5i−2:i∈1,t,ψ3′yi=125i−3:i∈1,2t−1,i∈ℕodd;125i:i∈2,2t,i∈ℕeven;ψ3′zi=125i−1:i∈1,2t−1,i∈ℕodd;125i−2:i∈2,2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence3,4,…,10t−1. ψ3′ is extendable to S−a,0−EAMT labeling of UP5t having a=15t (as per Lemma 1).(ii)
Forn=9:We construct a labelingψ3′:VUP9t⟶1,2,…,9t as follows:(20)ψ3″xi=9i−4:1≤i≤t,ψ3″yi=129i−3:i∈1,2t−1,i∈ℕodd;129i−4:i∈2,2t,i∈ℕeven;ψ3′zi=129i−7:i∈1,2t−1,i∈ℕodd;129i:i∈2,2t,i∈ℕeven;ψ3″wi=129i−5:i∈1,2t−1,i∈ℕodd;129i−2:i∈2,2t,i∈ℕeven;ψ3″vi=129i−1:i∈1,2t−1,i∈ℕodd;129i−6:i∈2,2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a natural numbers’ sequence3,4,…,18t−1. ψ3′′ is extendable to S−a,0−EAMT labeling of UP9t, by Lemma 1, with the admittance of a=27t.(iii)
Forn≥13:We are going to construct a labelingψ3:VUPnt⟶1,2,…,tn as follows:(21)ψ3xij=2ni−n+12:i∈1,t,j=1,ni−4j+52:i∈1,2t−1,j∈2,n+34,i∈ℕodd,ni+4j−n−32:i∈2,2t,j∈2,n+34,i∈ℕeven,ni+4j−2n−32:i∈1,2t−1,j∈n+74,n+12,i∈ℕodd,ni−4j+n+52:i∈2,2t,j∈n+74,n+12,i∈ℕeven.With the abovementioned scheme, the edge-sums being generated forms a consecutive integer sequence set3,4,…,2tn−1. Once again, ψ3 extends to a S−a,0−EAMT labeling of UPnt,n≥13 having a=3tn by Lemma 1.Forn≡3mod4:(i)
Forn=3:We define a labelingψ4′:VUP3t⟶1,2,…,3t as follows:(22)ψ4′xi=3i−2:1≤i≤t−1,i∈ℕodd,3i−1:2≤i≤t,i∈ℕeven,ψ4′yi=3i:1≤i≤t−1,i∈ℕodd,3i−2:2≤i≤t,i∈ℕeven,ψ4′zi=3i−1:1≤i≤t−1,i∈ℕodd,3i:2≤i≤t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers3,4,…,6t−1. Now, ψ4′ is extendable to S−a,0−EAMT labeling of UP3t≅Pt×C3 [15] with a=9t according to Lemma 1.(ii)
Forn=7:A labelingψ4″:VUP7t⟶1,2,…,7t is being defined as follows:(23)ψ4″xi=7i−3:1≤i≤t,ψ4″yi=127i−3:i∈1,2t−1,i∈ℕodd,127i−2:i∈2,2t,i∈ℕeven,ψ4″zi=127i−5:i∈1,2t−1,i∈ℕodd,127i:i∈2,2t,i∈ℕeven,ψ4″wi=127i−1:i∈1,2t−1,i∈ℕodd,127i−4:i∈2,2t,i∈ℕeven.The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence3,4,…,14t−1. Now, ψ4′′ is extendable to S−a,0−EAMT labeling of UP7t, according to Lemma 1 having a=21t.(iii)
Forn≥11:The labelingψ4:VUPnt⟶1,2,…,tn is constructed as follows:(24)ψ4xij=2ni−n+12:i∈1,t,j=1,ni−4j+52:i∈1,2t−1,j∈2,n+14,i∈ℕodd,ni+4j−n−32:i∈2,2t,j∈2,n+14,i∈ℕeven,ni+4j−2n−32:i∈1,2t−1,j∈n+54,n+12,i∈ℕodd,ni−4j+n+52:i∈2,2t,j∈n+54,n+12,i∈ℕeven.With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence set3,4,…,2tn−1. ψ4 is extendable to an S−a,0−EAMT labeling of UPnt,n≥11, under the light of Lemma 1, admitting a=3tn.
A direct outcome of Lemma2 is as follows.Theorem 3.
The pancyclic graphUPnt is S−tn+4,2−EAMT, for all t and n.


## 2.2.S−a,0−EAMT Labeling of Disjoint Union of C4 with Trees

It is a well-known fact that the graphC4 is not S−a,0−EAMT [6], and work is still in progress in order to determine if its disjoint copies are S−a,0−EAMT. In this section, we shall provide an S−a,0−EAMT labeling of disjoint copies of C4 with various trees in the form of several results. This will give a support to researchers to carry out their work to determine the aforesaid labeling of the disjoint copies of C4. Throughout this section, the union will represent a disjoint union of graphs only.Theorem 4.
For oddm, the graph mC4∪2K1,m∪7m−3/2K1 acquires an S−a,0−EAMT labeling admitting a=21m+2.Proof.
Consider a graphmC4∪2K1,m∪7m−3/2K1 with vertex and edge sets:(25)VΛ1=x1i,x2i:i∈1,m∪yi,zi:i∈1,m∪ki:1≤i≤2m∪li:1≤i≤7m−32∪c1,c2,EΛ1=yix1i,yix2i:1≤i≤m∪zix1i,zix2i:1≤i≤m∪c1ki:1≤i≤m∪c2ki:m+1≤i≤2m.
Ifp=VΛ1=19m+1/2 and q=EΛ1=6m, we sketch a labeling f1:VΛ1⟶1,2,…,19m+1/2 as follows:(26)f1x1i=127m+i,1≤i≤m;i≡1,mod2,126m+i,2≤i≤m−1;i≡0,mod2,f1x2i=5m−i−1:1≤i≤m,f1yi=i+4m+12,1≤i≤m;i≡1,mod2,i+5m+12,2≤i≤m−1;i≡0,mod2,f1zi=12i+10m+1,i∈1,m;i∈ℕodd,12i+11m+1,i∈2,m−1;i∈ℕeven,f1ki=i:i∈1,2m,f1c1=15m+12,f1c2=19m+12,f1li=6m+i,1≤i≤3m−12,i+6m+1,i∈3m+12,7m−32.
The edge-sums’ sets constituted by the abovementioned design generates a consecutive sequence of integer11m+3/2,11m+5/2,…,23m+1/2. Under the shadow of Lemma 1, f1 accredits an S−a,0−EAMT labeling of Λ1 having a=21m+2.Theorem 5.
For oddm, the graph mC4∪2mK2∪2mK1 acquires an S−a,0−EAMT labeling having a=43m+3/2.Proof.
Consider the graphΛ2≅mC4∪2mK2∪2mK1, for odd m, with the following vertex-edge connections:(27)VΛ2=x1i,x2i:i∈1,m∪zi,yi:1≤i≤m∪qi,pi:i∈1,2m∪li:i∈1,2m,EΛ2=yix1i,yix2i:1≤i≤m∪zix1i,zix2i:1≤i≤m∪qipi:i∈1,2m.
Here, order isp=10m and size is q=6m. Now, we design a labeling f2:VΛ2⟶1,2,…,10m as follows:(28)f2x1i=127m+i,i∈1,m;i∈ℕodd,126m+i,i≤2,m−1;i∈ℕeven,f2x2i=5m−i−1:i∈1,m,f2yi=12i+4m+1,i∈1,…,m;i∈ℕodd,12i+5m+1,i∈2,…,m−1;i∈ℕeven,f2zi=12i+10m+1,i∈1,m;i∈ℕodd,12i+11m+1,i∈2,m−1;i∈ℕeven,f2pi=i:i∈1,2m,f2qi=1215m−i+2:i∈1,m;i∈ℕodd,1216m−i+2:i∈2,m−1;i∈ℕeven,1221m−i+2:i∈m+2,2m−1;i∈ℕodd,1220m−i+2:i∈m+1,2m;i∈ℕeven,f2li=i+6m:i∈1,m,i+7m:i∈m+1,2m.
The edge-sums’ set constituted by the schemef2 generates a sequence consisting of consecutive integer 11m+3/2,11m+5/2,…,23m+1/2. Under the shadow of Lemma 1, f2 constitutes an S−a,0−EAMT labeling of Λ2 with magic sum a=43m+3/2.Theorem 6.
For oddm, the graph mC4∪2Pm+1∪5m−1/2K1 acquires an S−a,0−EAMT labeling having a=18m+5.Proof.
LetΛ3≅mC4∪2Pm+1∪5m−1/2K1, where m is odd, having vertex and edge sets interlinked:(29)VΛ3=x1i,x2i:i∈1,m∪zi,yi:i∈1,m∪pi:i∈1,m+1∪qi:i∈1,m+1∪li:i∈1,5m−12,EΛ3=yix1i,yix2i:i∈1,m∪zix1i,zix2i:i∈1,m∪pipi+1:i∈1,m∪qiqi+1:i∈1,m.
We havep=17m+3/2 and q=6m. A labeling f3:VΛ3⟶1,2,…,17m+3/2 is being defined as follows:(30)f3x1i=125m+i+2,i∈1,m;i∈ℕodd,124m+i+2,i∈2,m−1;i∈ℕeven,f3x2i=4m−i+2:1≤i≤m,f3yi=2m+i+32,i∈1,m;i∈ℕodd,3m+i+32,i∈2,m−1;i∈ℕeven,f3zi=8m+i+32,i∈1,…,m;i∈ℕodd,9m+i+32,i∈2,…,m−1;i∈ℕeven,f3pi=i+12,i∈1,m;i∈ℕodd,11m+i+32,i∈2,m+1;i∈ℕeven,f3qi=12i+m+2,i∈1,m;i∈ℕodd,1216m+i+2,i∈2,m+1;i∈ℕeven,f3li=i+5m+1:1≤i≤m+12,2i+11m+32:m+32≤i≤5m−12.
The edge-sums’ set constituted by the schemef3 generates a sequence consisting of consecutive integer 7m+7/2,7m+9/2,…,19m+5/2. Under the shadow of Lemma 1, f3 constitutes an S−a,0−EAMT of the graph Λ3 with magic constant a=18m+5.Theorem 7.
For oddm, mC4∪2m−2K2∪P4∪2mK1 possesses S−a,0−EAMT labeling having a=43m+5/2.Proof.
Withm taken odd, consider Λ4≅mC4∪2m−2K2∪P4∪2mK1, having vertex-edge connections:(31)VΛ4=x1i,x2i:i∈1,m∪yi,zi:i∈1,m∪li:1≤i≤2m∪qi,pi:i∈1,2m−1∪ti:i∈1,4,EΛ4=yix1i,yix2i:1≤i≤m∪zix1i,zix2i:i∈1,m∪piqi:i∈1,2m−1∪titi+1:1≤i≤3.
Here, we havep=10m and q=6m+1. A labeling function f:VΛ4⟶1,2,…,10m is being defined as follows:(32)f4x1i=127m+i:i∈1,m;i∈ℕodd,126m+i:i∈2,m−1;i∈ℕeven,f4x2i=5m−i−1:1≤i≤m,f4yi=i+4m+12:i∈1,…,m;i∈ℕodd,i+5m+12:i∈2,…,m−1;i∈ℕeven,f4zi=i+10m+12:i∈1,…,m;i∈ℕodd,i+11m+12:i∈2,…,m−1;i∈ℕeven,f4pi=i:i∈1,2m−1,f4qi=15m−i+22:i∈1,m;i∈ℕodd;16m−i+22:i∈2,m−1;i∈ℕeven;21m−i+22:i∈m+2,2m−3;i∈ℕodd;20m−i+22:iim+1,…,2m−1;i∈ℕeven;f4ti=2m−1:i=1;19m+32:i=2;2m:i=3;9m+1:i=4;f4li=i+6m:1≤i≤m;i+7m:i∈m+1,2m.
The edge-sums’ set constituted by the schemef4 generates a sequence consisting of consecutive integer 11m+3/2,11m+5/2,…,23m+3/2. Lemma 1 implies that f4 extends to an S−a,0−EAMT labeling of Λ4 with a=43m+5/2.Theorem 8.
For oddm, mC4∪Sm−1,m∪5m−1/2K1 possesses an S−a,0−EAMT labeling with a=29m+1.Proof.
Consider the graphΛ5≅mC4∪∪Sm−1,m∪5m−1/2K1 with vertex-edge connections as follows:(33)VΛ5=x1i,x2i:i∈1,m∪zi,yi:i∈1,m∪si,ti:i∈1,m∪li:i∈1,5m−12∪c,EΛ5=yix1i,yix2i:i∈1,m∪zix1i,zix2i:i∈1,m∪cti:i∈1,m∪tmsi:i∈1,m.
Then,p=17m+1/2 and q=6m. Again, a labeling f5:VΛ5⟶1,2,…,17m+1/2 is being defined as follows:(34)f5x1i=125m+i:i∈1,…,m;i∈ℕodd;124m+i:i∈2,m−1;i∈ℕeven,f5x2i=4m−i−1:i∈1,m,f5yi=i+2m+12:i∈1,…,m;i∈ℕodd;i+3m+12:i∈2,…,m−1;i∈ℕeven,f5zi=i+8m+12:i∈1,m;i∈ℕodd;i+9m+12:i∈2,m−1;i∈ℕeven,f5ti=i:1≤i≤m;f5si=2i+15m+12:1≤i≤m;fc=11m+12,f5li=i+5m:i∈1,m−12;i+5m+1:i∈m+12,5m−12.
The edge-sums’ set constituted by the schemef2 generates a sequence consisting of consecutive integer 7m+3/2,7m+5/2,…,19m+1/2. Under the shadow of Lemma 1, f5 constitutes to an S−a,0−EAMT labeling of Λ1 admitting a=29m+1.
The following results are direct consequences of Lemma2, from Theorems 4–8.Theorem 9.
For oddm, the graph mC4∪2K1,m∪7m−3/2K1 admits an S−15m+3,2−EAMT labeling.Theorem 10.
For oddm, the graph mC4∪2mP2∪2mK1 admits an S−31m+5/2,2−EAMT labeling.Theorem 11.
For oddm, the graph mC4∪2Pm+1∪5m−1/2K1 admits an S−12m+6,2−EAMT labeling.Theorem 12.
For oddm, mC4∪2m−2P2∪P4∪2mK1 admits an S−31m+5/2,2−EAMT labeling.Theorem 13.
For oddm, mC4∪Sm−1,m∪5m−1/2K1 admits an S−12m+3,2−EAMT labeling.


## 2.3. Examples and Proposed Open Problems

AnS−126,0−EAMT labeling of the graph UPnt is being presented in Figure 1(a), corresponding to the parameters t=3 and n=14. Furthermore, Figure 1(b) presents an S−96,0−EAMT labeling of UPnt corresponding to t=4 and n=8. Here, it can be observed that the value of the magic constant is perfect as per our depiction in Theorem 1.Figure 1
AnS−126,0−EAMT and S−96,0−EAMT labeling of the pancyclic graphs (a)UP143 and (b)UP84.
(a)(b)Figures2(a) and 2(b) illustrate Theorem 2 by providing S−117,0−EAMT and S−84,0−EAMT labeling of the graph UPnt.Figure 2
AnS−117,0−EAMT and S−84,0−EAMT labeling of the pancyclic graphs (a)UP133 and (b)UP74.
(a)(b)Figures3–7 are the illustrations of Theorems 4–8, respectively, for particular values of the parameters involved.Figure 3
AnS−107,0−EAMT labeling of the graph 5C4∪2K1,5∪16K1.Figure 4
AnS−109,0−EAMT labeling of the graph 5C4∪10K2∪10K1.Figure 5
AnS−95,0−EAMT labeling of the graph 5C4∪2P6∪12K1.Figure 6
AnS−109,0−EAMT labeling of the graph 5C4∪8K2∪P4∪10K1.Figure 7
AnS−92,0−EAMT labeling of the graph 5C4∪S4,5∪12K1.The open problems related to Section2.2 are as follows:(i)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2K1,m∪7m−3/2K1(ii)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2mK2∪2mK1(iii)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2Pm+1∪5m−1/2K1(iv)
For evenm, determine any S−a,0−EAMT labeling of mC4∪2m−2K2∪P4∪2mK1(v)
For evenm, determine any S−a,0−EAMT labeling of mC4∪Sm−1,m∪5m−1/2K1


## 3. Conclusion

In this article,(i)
We have obtainedS−a,0−EAMT and S−a′,2−EAMT labeling of a pancyclic class of graphs, namely, Usmanian pancyclic graph, denoted by UPnt.(ii)
We have exhibited the existence ofS−a,0−EAMT and S−a′,2−EAMT labeling on disjoint copies of C4 with various trees. Specifically, mC4∪2K1,m∪7m−3/2K1,mC4∪2mK2∪2mK1,mC4∪2Pm+1∪5m−1/2K1,mC4∪2m−2K2∪P4∪2mK1 and mC4∪Sm−1,m∪5m−1/2K1, whereas C4 itself is not S−a,0−EAMT. These obtained results open a new direction for researchers to derive S−a,0−EAMT labeling of disjoint copies of C4.(iii)
A few open problems have also been proposed for future work in this area.



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