# Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Relationship between $\mathbf{es}\mathbf{\left(}\mathit{G}\mathbf{\right)}$ and $\mathbf{mes}\mathbf{\left(}\mathit{G}\mathbf{\right)}$

**Theorem**

**1**

**.**Let G be a simple graph with $\mathrm{es}\left(G\right)=k$. If edge weights under a corresponding edge irregular k-labeling constitute a set of consecutive integers, then

**Theorem**

**2.**

**Proof.**

## 3. Fan Graphs

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

- Case (i). If $\phi \left(u\right)=1$, i.e., $\{\phi \left({v}_{i}\right):1\le i\le n\}=\{2,3,\dots ,n+1\}$, then the weights of edges ${v}_{i}u$, $1\le i\le n$, receive consecutive values from the set ${A}_{1}=\{3,4,\dots ,n+2\}$ and the weights of edges ${v}_{i}{v}_{i+1}$, $1\le i\le n-1$, attain values from the set ${A}_{2}=\{n+3,n+4,\dots ,2n+1\}$. The sum of the numbers in the set ${A}_{2}$ equals to the sum of the corresponding end vertex labels of edges, ${v}_{i}{v}_{i+1}$, $1\le i\le n-1$. The labels of vertices ${v}_{1}$ and ${v}_{n}$ are only counted once, while the labels of the vertices ${v}_{2},{v}_{3},\dots ,{v}_{n-1}$ are counted twice. We obtain the following:$$\begin{array}{c}\hfill 2\sum _{i=1}^{n}\phi \left({v}_{i}\right)-\phi \left({v}_{1}\right)-\phi \left({v}_{n}\right)=\sum _{i=1}^{n-1}w{t}_{\phi}\left({v}_{i}{v}_{i+1}\right),\end{array}$$$$\begin{array}{c}\hfill 2(2+3+\dots +(n+1))-\phi \left({v}_{1}\right)-\phi \left({v}_{n}\right)=(n+3)+(n+4)+\dots +(2n+1)\end{array}$$$$\begin{array}{c}\hfill \phi \left({v}_{1}\right)+\phi \left({v}_{n}\right)={\textstyle \frac{-{n}^{2}+5n+4}{2}}.\end{array}$$$$\begin{array}{c}\hfill 5\le {\textstyle \frac{-{n}^{2}+5n+4}{2}}\le 2n+1.\end{array}$$$$(n-3)(n-2)\le 0\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}(n-2)(n+1)\ge 0,$$
- Case (ii). If $\phi \left(u\right)=n+1$, i.e., $\{\phi \left({v}_{i}\right):1\le i\le n\}=\{1,2,\dots ,n\}$, then by Lemma 2 this case is analogous to Case (i).
- Case (iii). Assume $\phi \left(u\right)=s$, $1<s<n+1$. Now, the set of labels of vertices ${v}_{1},{v}_{2},\dots ,{v}_{n}$ consists of two subsets $C=\{1,2,\dots ,s-1\}$ and $D=\{s+1,s+2,\dots ,n+1\}$. Then, corresponding weights of edges ${v}_{i}u$ form the set $W=\{wt\left({v}_{i}u\right):1\le i\le n\}=\{s+1,s+2,\dots ,2s-2,2s-1,2s+1,2s+2,\dots ,s+n,s+n+1\}$.

**Corollary**

**1.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**2.**

## 4. Wheels

**Lemma**

**3.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

## 5. Conclusions

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ahmad, A.; Al-Mushayt, O.; Bača, M. On edge irregularity strength of graphs. Appl. Math. Comput.
**2014**, 243, 607–610. [Google Scholar] [CrossRef] - Chartrand, G.; Jacobson, M.S.; Lehel, J.; Oellermann, O.R.; Ruiz, S.; Saba, F. Irregular networks. Congr. Numer.
**1988**, 64, 187–192. [Google Scholar] - Ahmad, A.; Bača, M.; Nadeem, M.F. On edge irregularity strength of Toeplitz graphs. U.P.B. Sci. Bull. Ser. A
**2016**, 78, 155–162. [Google Scholar] - Tarawneh, I.; Hasni, R.; Ahmad, A. On the edge irregularity strength of grid graphs. AKCE Int. J. Graphs Comb.
**2020**, 17, 414–418. [Google Scholar] [CrossRef] - Tarawneh, I.; Hasni, R.; Ahmad, A.; Asim, M.A. On the edge irregularity strength for some classes of plane graphs. AIMS Math.
**2021**, 6, 2724–2731. [Google Scholar] [CrossRef] - Koam, A.N.A.; Ahmad, A.; Bača, M.; Semaničová-Feňovčíková, A. Modular edge irregularity strength of graphs. AIMS Math.
**2023**, 8, 1475–1487. [Google Scholar] [CrossRef] - Bača, M.; Muthugurupackiam, K.; Kathiresan, K.M.; Ramya, S. Modular irregularity strength of graphs. Electron. J. Graph Theory Appl.
**2020**, 8, 435–443. [Google Scholar] [CrossRef] - Muthugurupackiam, K.; Ramya, S. Modular irregularity strength of graphs. J. Comput. Math. Sci.
**2018**, 9, 1132–1141. [Google Scholar] - Bača, M.; Kimáková, Z.; Lascsáková, M.; Semaničová-Feňovčíková, A. The irregularity and modular irregularity strength of fan graphs. Symmetry
**2021**, 13, 605. [Google Scholar] [CrossRef] - Sugeng, K.A.; Barack, Z.Z.; Hinding, N.; Simanjuntak, R. Modular irregular labeling on double-star and friendship graphs. J. Math.
**2021**, 2021, 4746609. [Google Scholar] [CrossRef] - Tilukay, M.I. Modular irregularity strength of triangular book graph. Tensor-Pure Appl. Math. J.
**2021**, 2, 53–58. [Google Scholar] - Nisa, I.C. Modular irregular labeling on complete graphs. Daya-Mat.-J. Inov. Pendidik. Mat.
**2022**, 10. [Google Scholar] [CrossRef] - Ahmad, A.; Gupta, A.; Simanjuntak, R. Computing the edge irregularity strengths of chain graphs and the join of two graphs. Electron. J. Graph Theory Appl.
**2018**, 6, 201–207. [Google Scholar] [CrossRef] [Green Version] - Hartsfield, N.; Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction; Academic Press: Boston, DC, USA; San Diego, CA, USA; New York, NY, USA; London, UK, 1990. [Google Scholar]

**Figure 2.**The edge irregular $(n+1)$-labelings of ${F}_{n}$ for $(n,s)=(4,3)$, $(n,s)=(5,3)$, $(n,s)=(5,4)$ and $(n,s)=(6,4)$.

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**MDPI and ACS Style**

Haryeni, D.O.; Awanis, Z.Y.; Bača, M.; Semaničová-Feňovčíková, A.
Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs. *Symmetry* **2022**, *14*, 2671.
https://doi.org/10.3390/sym14122671

**AMA Style**

Haryeni DO, Awanis ZY, Bača M, Semaničová-Feňovčíková A.
Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs. *Symmetry*. 2022; 14(12):2671.
https://doi.org/10.3390/sym14122671

**Chicago/Turabian Style**

Haryeni, Debi Oktia, Zata Yumni Awanis, Martin Bača, and Andrea Semaničová-Feňovčíková.
2022. "Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs" *Symmetry* 14, no. 12: 2671.
https://doi.org/10.3390/sym14122671