# On the Metric Dimension of Arithmetic Graph of a Composite Number

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Results

**Proposition**

**1.**

**Proposition**

**2.**

**Theorem**

**1.**

- (i)
- $diam\left({\mathcal{A}}_{m}\right)=2$ if and only if ${\displaystyle \sum}_{i=1}^{t}}{\gamma}_{i}=t$.
- (ii)
- $diam\left({\mathcal{A}}_{m}\right)=3$ if and only if ${\displaystyle \sum}_{i=1}^{t}}{\gamma}_{i}\ge t+1$.

**Proof.**

**Proposition**

**3.**

**Proof.**

**Lemma**

**1.**

**Proof.**

#### Metric Dimension of Arithmetic Graphs

**Corollary**

**1.**

**Theorem**

**2.**

- (i)
- For ${\gamma}_{1}=1$ and ${\gamma}_{2}=2$, $dim\left({\mathcal{A}}_{m}\right)=2$.
- (ii)
- For ${\gamma}_{1}=1$ and ${\gamma}_{2}\ge 3$, $dim\left({\mathcal{A}}_{m}\right)=2{\gamma}_{2}-3$.
- (iii)
- For ${\gamma}_{1}=2$ and ${\gamma}_{2}=2$, $dim\left({\mathcal{A}}_{m}\right)=3$.
- (iv)
- For ${\gamma}_{1}=2$ and ${\gamma}_{2}\ge 3$, $dim\left({\mathcal{A}}_{m}\right)=2+3({\gamma}_{2}-2)$.
- (v)
- For ${\gamma}_{1},{\gamma}_{2}\ge 3$, $dim\left({\mathcal{A}}_{m}\right)=1+2({\gamma}_{1}-2)+2({\gamma}_{2}-2)+(({\gamma}_{1}-1)({\gamma}_{2}-1)-1)$.

**Proof.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${\mathcal{A}}_{m}$ | Arithmetic graph of a composite number m with at least two distinct primary divisors |

$diam\left(G\right)$ | The diameter of a graph G |

$dim\left(G\right)$ | The metric dimension of a graph G |

$deg\left(v\right)$ | The degree of a vertex v |

$N\left(v\right)$ | The open neighborhood of a vertex v |

$d(x,y)$ | The distance between the vertices x and y |

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**MDPI and ACS Style**

Rehman, S.u.; Imran, M.; Javaid, I.
On the Metric Dimension of Arithmetic Graph of a Composite Number. *Symmetry* **2020**, *12*, 607.
https://doi.org/10.3390/sym12040607

**AMA Style**

Rehman Su, Imran M, Javaid I.
On the Metric Dimension of Arithmetic Graph of a Composite Number. *Symmetry*. 2020; 12(4):607.
https://doi.org/10.3390/sym12040607

**Chicago/Turabian Style**

Rehman, Shahid ur, Muhammad Imran, and Imran Javaid.
2020. "On the Metric Dimension of Arithmetic Graph of a Composite Number" *Symmetry* 12, no. 4: 607.
https://doi.org/10.3390/sym12040607