# Distance Antimagic Product Graphs

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## Abstract

**:**

## 1. Introduction

**Conjecture**

**1**

**.**A graph G is distance antimagic if and only if G does not have two vertices with the same open neighborhood.

## 2. Graph Products: Definition and Notation

**Definition**

**1.**

**Cartesian product of G and H**, denoted by $G\square H$, is the graph with $V(G\square H)=V(G)\times V(H)$ and two vertices $(u,{u}^{\prime})$ and $(v,{v}^{\prime})$ are adjacent if and only if either

- 1.
- $u=v$ and ${u}^{\prime}$ is adjacent to ${v}^{\prime}$ in H, or
- 2.
- ${u}^{\prime}={v}^{\prime}$and u is adjacent to v in G.

**Definition**

**2.**

**direct product of G and H**, denoted by $G\times H$, is the graph with $V(G\times H)=V(G)\times V(H)$ and the two vertices $(u,{u}^{\prime})$ and $(v,{v}^{\prime})$ are adjacent if and only if u is adjacent to v and ${u}^{\prime}$ is adjacent to ${v}^{\prime}$.

**Definition**

**3.**

**strong product of G and H**, denoted by $G\u22a0H$, is the graph with $V(G\u22a0H)=V(G)\times V(H)$, and the two vertices $(u,{u}^{\prime})$ and $(v,{v}^{\prime})$ are adjacent if and only if either

- 1.
- u is adjacent to v, and ${u}^{\prime}$ is adjacent to ${v}^{\prime}$, or
- 2.
- $u=v$ and ${u}^{\prime}$ is adjacent to ${v}^{\prime}$ in H, or
- 3.
- ${u}^{\prime}={v}^{\prime}$ and u is adjacent to v in G.

**Definition**

**4.**

**lexicographic product of graphs G and H**, denoted by $G\circ H$, is a graph with $V(G\circ H)=V(G)\times V(H)$ and the two vertices $(u,{u}^{\prime})$ and $(v,{v}^{\prime})$ are adjacent if and only if either

- 1.
- $u=v$ and ${u}^{\prime}$ is adjacent to ${v}^{\prime}$ in H, or
- 2.
- u and v are adjacent in G.

**Definition**

**5**

**.**The

**corona product of G and H**, denoted by $G\odot H$, is the graph obtained by taking a copy of G and $|V(G)|$ copies of H and joining the i-th vertex of G to every vertex in the i-th copy of H.

**Definition**

**6.**

**monotone**if there exists a vertex labeling λ, i.e., a bijection $\lambda :V(G)\to \{1,2,\dots ,n\}$, such that $\lambda (u)<\lambda (v)$ implies $\omega (u)\le \omega (v)$ for every pair of distinct vertices $u,v$ in G.

## 3. Distance Antimagic Graphs Obtained from Fundamental Graph Products

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Problem**

**1.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 4. Distance Antimagic Graphs Obtained from the Lexicographic Product

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Definition**

**7.**

**${G}_{a}$**be the subgraph of $H\circ G$ induced by $\{(a,v)|v\in V(G)\}$. Define a labeling λ for $H\circ G$ by $\lambda (a,v)={\lambda}_{G}(v)+{\lambda}_{H}(a)n$ for $(a,v)\in V(H\circ G)$.

**Lemma**

**3.**

**Proof.**

**Definition**

**8.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 5. Distance Antimagic Graphs Obtained from the Corona Product

**Corollary**

**1.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Problem**

**2.**

**Theorem**

**14.**

**Proof.**

**Theorem**

**15.**

**Proof.**

**Problem**

**3.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Examples of fundamental graph products: ${P}_{3}\square {P}_{4}$, ${P}_{3}\times {P}_{4}$, and ${P}_{3}\u22a0{P}_{4}$.

**Figure 4.**Examples of distance antimagic labeling for ${P}_{n}\square {K}_{2},\text{}n=3,4,5$. The vertices’ labels are written in black, while their weights are blue.

**Figure 5.**The bipartition sets of $V({K}_{n,n})$ (

**left**) and the product graph ${K}_{n,n}\square {K}_{2}$ (

**right**).

**Figure 6.**Examples of distance antimagic labeling for ${P}_{n}\square {K}_{3},\text{}n=6,7,9$. The vertices’ labels are written in black, while their weights are blue.

**Figure 7.**Graph ${C}_{4}$ with its distance antimagic labeling (

**left**) and graph ${C}_{4}\circ G$ with its induced subgraphs ${G}_{a}$s (

**right**).

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**MDPI and ACS Style**

Simanjuntak, R.; Tritama, A.
Distance Antimagic Product Graphs. *Symmetry* **2022**, *14*, 1411.
https://doi.org/10.3390/sym14071411

**AMA Style**

Simanjuntak R, Tritama A.
Distance Antimagic Product Graphs. *Symmetry*. 2022; 14(7):1411.
https://doi.org/10.3390/sym14071411

**Chicago/Turabian Style**

Simanjuntak, Rinovia, and Aholiab Tritama.
2022. "Distance Antimagic Product Graphs" *Symmetry* 14, no. 7: 1411.
https://doi.org/10.3390/sym14071411