# On Edge Irregular Reflexive Labeling for Generalized Prism

Authors: Chenxi Wang; M. J. A. Khan; M. Ibrahim; E. Bonyah; M. K. Siddiqui; S. Khalid
Journal: Journal of Mathematics
 (2022)

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## Abstract

Among the various ideas that appear while studying graph theory, which has gained much attraction especially in graph labeling, labeling of graphs gives mathematical models which value for a vast range of applications in high technology (data security, cryptography, various problems of coding theory, astronomy, data security, telecommunication networks, etc.). A graph label is a designation of graph elements, i.e., the edges and/or vertex of a group of numbers (natural numbers), and is called assignment or labeling. The vertex or edge labeling is related to their domain asset of vertices or edges. Likewise, for total labeling, we take the domain as vertices and edges both at the same time. The reflexive edge irregularity strength (res) is total labeling in which weights of edges are not the same for all edges and the weight of an edge is taken as the sum of the edge labels and the vertices associated with that edge. In the res, the vertices are labeled with nonnegative even integers while the edges are labeled with positive integers. We have to make the labels minimum, whether they are associated with vertices or edges. If such labeling exists, then it is called the res ofH and is represented as sresH. In this paper, we have computed the res for the Cartesian product of path and cycle graph which is also known as generalizing prism.


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## Body

## 1. Introduction

Any graphH is the combination of vertices VH along with a possibly nonempty edge set EH of 2−element subsets of VH. In this paper, all the chosen graphs are finite, without direction, nontrivial, connected, and simple (without loops and multiedges). For details about notations, see [1, 2]. Nonnegative integers are used in this research. In 1988, Chartrand et al. [3] proposed the labeling problems in graph theory. Assign the edges positive integer to all connected simple graphs such as the graph became irregular. The irregular labeling is defined as ψ:EH⟶1,2,3,…,m and is called irregular m−labeling for graph H if all the separate nodes u and u′ have distinctly weights, that is,(1)∑x∈Vψux≠∑y∈Vψu′y.Lahel, in [4], studied, in detail, for the irregularity strength. For more results, see the works of Nierhoff in [5], Dimitz et al. in [6], Amar and Togni in [7], and Gyarfas in [8].In [9], A. Ahmad et al. defined on edge irregularity strength esH for any two edges u1u2 and u1′u2′ that the weights wϕu1u2 and wϕu1′u2′ are distinct, as weight for an edge u1u2∈EH is wϕu1u2=ϕu1+ϕu2.In [10], Bača et al. defined the parameter of total labeling for edge as well as vertex of graph and found the weights of an edge as sum of three integers which include the edge label and the labels of two vertices associated with that edge, and finally, every edge has distinct weight. For detailed studies on total edge irregularity strength, see [10, 11].The concept of total edge irregularity strength has been generalized by Zhang et al. in [12] for graph will be reflexive edge irregularity strength m−labeling.If, for any graphH, the total m−labeling defined the mapping ψe′:EH⟶1,2,3…,me′ and ψv′:VH⟶0,2,4,…,2mv′, the mapping ψ is a total m−mapping of H such that ψa=ψva if a∈VH and ψa=ψea if a∈EH, where k=maxme′,2mv′.The totalp−labeling ψ will be edge irregular reflexive p−labeling of the graph H if, for all the different edges say u1u2 and u1′u2′, the weights wϕu1u2 and wϕu1′u2′ are not the same for every choice of edges where the weight for any edge suppose u1u2∈EH is wϕu1u2=ϕu1+ϕu1u2+ϕu2.The smallest value ofp for which such mapping exists is said to be res of the graph H and is represented by resH. For details in reflexive edge irregularity strength, see [13–17].ForresH, Nierhoff [5] proposed that for any graph Hs,t with maximum degree ΔH satisfies(2)resH=maxt3+r,Δ2+1,where r will be 1 for t≡2,3mod6; it will be 0, otherwise.In [12], the lemma is proven.Lemma 1.
For all graph sayH,(3)resH≥t3+1,ift≡2,3mod6,t3,ift≡1,4,5mod6.In the present research paper, we have investigated the res for the Cartesian product of paths and cycles.


### 1.1. Definition

The Cartesian productP and Q graphs is represented as P□Q and is the graph with vertices set VP×VQ, with vertices u1,u1′ and w1,w1′ will be adjacent if and only if u1=w1 and u1′w1′∈EQ or u1′=w1′ and u1w1∈EP.Theorem 1.
LetPd and Cc be path and cycle, respectively; then, for edge irregular reflexive strength of Pd□Cc with d≥3 and c≥2. We have(4)resPd□Cc=2d−1c3+1,if2d−1c≡2,3mod6,2d−1c3,if2d−1c≡1,4,5,6mod6.Proof.
AsPd□Cc has 2d−1c edges, therefore, from Lemma 1, we obtain(5)resPd□Cc≥2d−1c3+1,if2d−1c≡2,3mod6,2d−1c3,if2d−1c≡1,4,5,6mod6.
Next, we will show that(6)resPd□Cc≤2d−1c3+1,if2d−1c≡2,3mod6,2d−1c3,if2d−1c≡1,4,5,6mod6.
We defined af−labeling for this on Pd□Cc as follows: ∀1≤j≤c.
Lete=xi,j, h=xi+1,j, and k=xi,j+1Case 1.
Whend≡0mod3, c is odd:(7)fe=ci−1,for1≤i≤2d−33iis odd,ci−1−1,for2≤i≤2d3iis even,k,for2d+33≤i≤d.
Whenc≡1mod6,(8)fek=j,for1≤i≤2d−33,j+2,for2≤i≤2d3,6ci−22cd−c−13+j,for2d+33≤i≤d,feh=j+1,for1≤i≤2d−33,c−13+1+jfori=2d3,6ci−4cd+c+23+j,for2d+33≤i≤d−1.
Whenc≡3mod6,(9)fek=j,for1≤i≤2d−33,j+2,for2≤i≤2d3,6ci−22cd+2c+33+jfor2d3+1≤i≤d,feh=j+1,for1≤i≤2d−33,2c−13+1+jfori=2d3,6ci−4cd+13+j,for2d+33≤i≤d−1.
Whenc≡5mod6,(10)fek=j,for1≤i≤2d−33iis odd,j+1,for2≤i≤2d3iis even,6ci−22cd+2d−13+j,for2d+33≤i≤d,feh=j+1,for1≤i≤2d3−1,c−53+3+jfori=2d3,6ci−4cd+c−23+j,for2d3+1≤i≤d−1.Case 2.
Whend≡0mod3, c is even:(11)fe=di−1,for1≤i≤2d3,k,for2d+33≤i≤d.
Whend≡0mod6,(12)fek=j,for1≤i≤2d3,6i−4d−4c3+j,for2d+33≤i≤d,feh=j,for1≤i≤2d3−1,c3+jfori=2d3,6i−4d−1c3+j,for2d3+1≤i≤d−1.
Whenc≡2mod6,(13)fek=j,for1≤i≤2d3,6ci−4cd+c+13+j,for2d+33≤i≤d,feh=j,for1≤i≤2d−33,c−23+jfori=2d3,6ci−4cd+c+43+j,for2d3+1≤i≤d−1.
Whenc≡4mod6,(14)fek=j,for1≤i≤2d3,6ci−4cd+c+23+j,for2d3+1≤i≤d,feh=j,for1≤i≤2d3−1,2c−43+jfori=2d3,6ci−4cd+c+83+j,for2d3+1≤i≤d−1.Case 3.
Whend≡1mod3, c is even:(15)fe=ci−1,for1≤i≤2d+13,k,for2d+23≤i≤d.
Whend≡1mod3 and c≡0mod6,(16)fek=j,for1≤i≤2d+13,c6i−4d−43+j,for2d+23≤i≤d,feh=j,for1≤i≤2d−13,2c3+j,fori=2d+13,c6i−4d−13+j,for2d+43≤i≤d−1.
Whend≡1mod3 and c≡2mod6,(17)fek=j,for1≤i≤2d+13,6ci−4cd+c+23+j,for2d+23≤i≤d,feh=j,for1≤i≤2d−13,2c−23+j,fori=2d+13,6ci−4cd+c+83+j,for2d+23≤i≤d−1.
Whend≡1mod3 and c≡4mod6,(18)fek=j,for1≤i≤2d+13,6ci−4cd+c+13+jfor2d+23≤i≤d,feh=j,for1≤i≤2d−13,2c−43+2+j,fori=2d+13,6ci−4cd+c+43+j,for2d+23≤i≤d−1.Case 4.
Whend≡1mod3, c is odd:(19)fe=ci−1,for1≤i≤2d+13iis odd.ci−1−1,for2≤i≤2d−13iis even.k,for2d+23≤i≤d.
An illustration of this reflexive labeling is shown in Figures1 and 2.
Whend≡1mod3 and c≡1mod6,(20)fek=j,for1≤i≤2d+13iis odd,j+2,for2≤i≤2n−13iis even,6ci−4cd+4c−23+jfor2d+23≤i≤d,feh=j+1,for1≤i≤2d−13,2c+13+j,fori=2d+13,6ci−4cd+53+j,for2n+23≤i≤d−1.
Whend≡1mod3 and c≡3mod6,(21)fek=j,for1≤i≤2d+13iis odd,j+2,for2≤i≤2d−13iis even,c6i−4d−43−2+jfor2d+23≤i≤d,feh=j+1,for1≤i≤2d−13,2c3−1+j,fori=2d+13,c6i−4d−13−2+j,for2d+23≤i≤d−1.
Whend≡1mod3 and c≡5mod6,(22)fek=j,for1≤i≤2d+13iis odd,j+2,for1≤i≤2d−13iis even,6ci−4cd+1−23+jfor2d+23≤i≤d,feh=j+1,for1≤i≤2d−13,2c−53+3+j,fori=2d+13,6ci−4cd+c+23+j,for2d+23≤i≤d−1.Figure 1
Twenty four labeling ofP4□C10.Figure 2
Edge weights ofP4□C10.Case 5.
Whend≡2mod3, c is even and c≥4:(23)fe=ci−1,for1≤i≤2d−13,k,for2d+23≤i≤d,fek=j,for1≤i≤2d+23,6ci−4cd−2c+33+jfor2d+53+3≤i≤d,feh=j,for1≤i≤2d−13,6ci−4d−2c−93+j,for2d+23≤i≤d−1.Case 6.
Whend≡2mod3, c is odd and c≥3:(24)fe=ci−1,for1≤i≤2d−13iis odd,ci−1−1,for2≤i≤2d+23iis even,k,for2d+53≤i≤d,fek=j,for1≤i≤2d−13iis odd,j+2,for2≤i≤2d+23iis even,6ci−4d+1c3−2+jfor2d+53≤i≤d,feh=j+1,for1≤i≤2d−13,2c−2i−4d+1c−13−1+j,for2d+23≤i≤d−1.
Weights for reflexive edges is given as follows:for1≤i≤d−11,≤i≤c, weight of edge xi,jxi+1,j is 2i−1c+j and weight of the edge xi,jxi,j+1 is 2ci−1+j,
None of the two edges are of the same weight. So, we get the required result, forc≥2andd≥3, which completes the proof.


## 1.1. Definition

The Cartesian productP and Q graphs is represented as P□Q and is the graph with vertices set VP×VQ, with vertices u1,u1′ and w1,w1′ will be adjacent if and only if u1=w1 and u1′w1′∈EQ or u1′=w1′ and u1w1∈EP.Theorem 1.
LetPd and Cc be path and cycle, respectively; then, for edge irregular reflexive strength of Pd□Cc with d≥3 and c≥2. We have(4)resPd□Cc=2d−1c3+1,if2d−1c≡2,3mod6,2d−1c3,if2d−1c≡1,4,5,6mod6.Proof.
AsPd□Cc has 2d−1c edges, therefore, from Lemma 1, we obtain(5)resPd□Cc≥2d−1c3+1,if2d−1c≡2,3mod6,2d−1c3,if2d−1c≡1,4,5,6mod6.
Next, we will show that(6)resPd□Cc≤2d−1c3+1,if2d−1c≡2,3mod6,2d−1c3,if2d−1c≡1,4,5,6mod6.
We defined af−labeling for this on Pd□Cc as follows: ∀1≤j≤c.
Lete=xi,j, h=xi+1,j, and k=xi,j+1Case 1.
Whend≡0mod3, c is odd:(7)fe=ci−1,for1≤i≤2d−33iis odd,ci−1−1,for2≤i≤2d3iis even,k,for2d+33≤i≤d.
Whenc≡1mod6,(8)fek=j,for1≤i≤2d−33,j+2,for2≤i≤2d3,6ci−22cd−c−13+j,for2d+33≤i≤d,feh=j+1,for1≤i≤2d−33,c−13+1+jfori=2d3,6ci−4cd+c+23+j,for2d+33≤i≤d−1.
Whenc≡3mod6,(9)fek=j,for1≤i≤2d−33,j+2,for2≤i≤2d3,6ci−22cd+2c+33+jfor2d3+1≤i≤d,feh=j+1,for1≤i≤2d−33,2c−13+1+jfori=2d3,6ci−4cd+13+j,for2d+33≤i≤d−1.
Whenc≡5mod6,(10)fek=j,for1≤i≤2d−33iis odd,j+1,for2≤i≤2d3iis even,6ci−22cd+2d−13+j,for2d+33≤i≤d,feh=j+1,for1≤i≤2d3−1,c−53+3+jfori=2d3,6ci−4cd+c−23+j,for2d3+1≤i≤d−1.Case 2.
Whend≡0mod3, c is even:(11)fe=di−1,for1≤i≤2d3,k,for2d+33≤i≤d.
Whend≡0mod6,(12)fek=j,for1≤i≤2d3,6i−4d−4c3+j,for2d+33≤i≤d,feh=j,for1≤i≤2d3−1,c3+jfori=2d3,6i−4d−1c3+j,for2d3+1≤i≤d−1.
Whenc≡2mod6,(13)fek=j,for1≤i≤2d3,6ci−4cd+c+13+j,for2d+33≤i≤d,feh=j,for1≤i≤2d−33,c−23+jfori=2d3,6ci−4cd+c+43+j,for2d3+1≤i≤d−1.
Whenc≡4mod6,(14)fek=j,for1≤i≤2d3,6ci−4cd+c+23+j,for2d3+1≤i≤d,feh=j,for1≤i≤2d3−1,2c−43+jfori=2d3,6ci−4cd+c+83+j,for2d3+1≤i≤d−1.Case 3.
Whend≡1mod3, c is even:(15)fe=ci−1,for1≤i≤2d+13,k,for2d+23≤i≤d.
Whend≡1mod3 and c≡0mod6,(16)fek=j,for1≤i≤2d+13,c6i−4d−43+j,for2d+23≤i≤d,feh=j,for1≤i≤2d−13,2c3+j,fori=2d+13,c6i−4d−13+j,for2d+43≤i≤d−1.
Whend≡1mod3 and c≡2mod6,(17)fek=j,for1≤i≤2d+13,6ci−4cd+c+23+j,for2d+23≤i≤d,feh=j,for1≤i≤2d−13,2c−23+j,fori=2d+13,6ci−4cd+c+83+j,for2d+23≤i≤d−1.
Whend≡1mod3 and c≡4mod6,(18)fek=j,for1≤i≤2d+13,6ci−4cd+c+13+jfor2d+23≤i≤d,feh=j,for1≤i≤2d−13,2c−43+2+j,fori=2d+13,6ci−4cd+c+43+j,for2d+23≤i≤d−1.Case 4.
Whend≡1mod3, c is odd:(19)fe=ci−1,for1≤i≤2d+13iis odd.ci−1−1,for2≤i≤2d−13iis even.k,for2d+23≤i≤d.
An illustration of this reflexive labeling is shown in Figures1 and 2.
Whend≡1mod3 and c≡1mod6,(20)fek=j,for1≤i≤2d+13iis odd,j+2,for2≤i≤2n−13iis even,6ci−4cd+4c−23+jfor2d+23≤i≤d,feh=j+1,for1≤i≤2d−13,2c+13+j,fori=2d+13,6ci−4cd+53+j,for2n+23≤i≤d−1.
Whend≡1mod3 and c≡3mod6,(21)fek=j,for1≤i≤2d+13iis odd,j+2,for2≤i≤2d−13iis even,c6i−4d−43−2+jfor2d+23≤i≤d,feh=j+1,for1≤i≤2d−13,2c3−1+j,fori=2d+13,c6i−4d−13−2+j,for2d+23≤i≤d−1.
Whend≡1mod3 and c≡5mod6,(22)fek=j,for1≤i≤2d+13iis odd,j+2,for1≤i≤2d−13iis even,6ci−4cd+1−23+jfor2d+23≤i≤d,feh=j+1,for1≤i≤2d−13,2c−53+3+j,fori=2d+13,6ci−4cd+c+23+j,for2d+23≤i≤d−1.Figure 1
Twenty four labeling ofP4□C10.Figure 2
Edge weights ofP4□C10.Case 5.
Whend≡2mod3, c is even and c≥4:(23)fe=ci−1,for1≤i≤2d−13,k,for2d+23≤i≤d,fek=j,for1≤i≤2d+23,6ci−4cd−2c+33+jfor2d+53+3≤i≤d,feh=j,for1≤i≤2d−13,6ci−4d−2c−93+j,for2d+23≤i≤d−1.Case 6.
Whend≡2mod3, c is odd and c≥3:(24)fe=ci−1,for1≤i≤2d−13iis odd,ci−1−1,for2≤i≤2d+23iis even,k,for2d+53≤i≤d,fek=j,for1≤i≤2d−13iis odd,j+2,for2≤i≤2d+23iis even,6ci−4d+1c3−2+jfor2d+53≤i≤d,feh=j+1,for1≤i≤2d−13,2c−2i−4d+1c−13−1+j,for2d+23≤i≤d−1.
Weights for reflexive edges is given as follows:for1≤i≤d−11,≤i≤c, weight of edge xi,jxi+1,j is 2i−1c+j and weight of the edge xi,jxi,j+1 is 2ci−1+j,
None of the two edges are of the same weight. So, we get the required result, forc≥2andd≥3, which completes the proof.


## 2. Conclusion

In the present paper, we found the reflexive edge irregularity strength for generalized prism graphPd□Cc, for d≥3 and c≥2.



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