# Open Support of Hypergraphs under Addition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 3. Examples

**Theorem**

**1.**

**Proof.**

- For $i=1,m$, $1\le j\le k-1$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k$.
- For $2\le i\le m-1,1\le j\le k-2$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+1$.

**Example**

**1.**

**Theorem**

**2.**

**Proof.**

- For $1\le j\le k-1$, $supp\left({v}_{1}^{j}\right)={\sum}_{u\in N\left({v}_{1}^{j}\right)}d\left(u\right)=k+2$;
- For $2\le i\le m-1,1\le j\le k-2$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+5$;
- For $1\le j\le k-1$, $supp\left({v}_{m}^{j}\right)={\sum}_{u\in N\left({v}_{m}^{j}\right)}d\left(u\right)=k+2$;
- For $m+1\le i\le 3m-2$, $1\le j\le k-1$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+2$.

**Example**

**2.**

**Theorem**

**3.**

**Proof.**

- For $1\le i\le m,1\le j\le k-1$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+m$;
- For $m+1\le i\le m+n,1\le j\le k-1$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+n$;
- For $m+n+1\le i\le m+n+2$, $1\le j\le k-2$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+n+2$;
- For $m+n+3\le i\le m+n+4$, $1\le j\le k-2$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+m+2$;
- For $1\le j\le k-2$, $supp\left({v}_{m+n+5}^{j}\right)={\sum}_{u\in N\left({v}_{m+n+5}^{j}\right)}d\left(u\right)=k+3$.

**Example**

**3.**

**Theorem**

**4.**

**Proof.**

- For $1\le j\le k-1$, $supp\left({v}_{1}^{j}\right)={\sum}_{u\in N\left({v}_{1}^{j}\right)}d\left(u\right)=k;$
- For $2\le i\le m-1,1\le j\le k-2$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+1;$
- For $m\le i\le m+1$, $1\le j\le k-2$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+2m+1;$
- For $m+2\le i\le 2m-1,1\le j\le k-2$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+1;$
- For $1\le j\le k-1$, $supp\left({v}_{2m}^{j}\right)={\sum}_{u\in N\left({v}_{2m}^{j}\right)}d\left(u\right)=k;$
- For $2m+1\le i\le 4m,1\le j\le k-1$, $supp\left({v}_{i}^{j}\right)={\sum}_{u\in N\left({v}_{i}^{j}\right)}d\left(u\right)=k+2m.$

**Example**

**4.**

**Theorem**

**5.**

**Proof.**

- For $1\le q\le k-2$, $supp\left({v}_{1}^{q}\right)={\sum}_{u\in N\left({v}_{1}^{q}\right)}d\left(u\right)=k+3$;
- For $2\le i\le m-2,1\le q\le k-2$, $supp\left({v}_{i}^{q}\right)={\sum}_{u\in N\left({v}_{i}^{q}\right)}d\left(u\right)=2i+k+2$;
- For $1\le t\le k-2$, $supp\left({v}_{m-1}^{q}\right)={\sum}_{u\in N\left({v}_{m-1}^{q}\right)}d\left(u\right)=2m+k-1$;
- For $1\le q\le k-1$, $supp\left({v}_{1,1}^{q}\right)={\sum}_{u\in N\left({v}_{1,1}^{q}\right)}d\left(u\right)=k$;
- For $2\le i\le m-1,1\le j\le i,1\le t\le k-1$, $supp\left({v}_{i,j}^{q}\right)={\sum}_{u\in N\left({v}_{i,j}^{q}\right)}d\left(u\right)=k+i$;
- For $1\le j\le m,1\le q\le k-1$, $supp\left({v}_{m,j}^{q}\right)={\sum}_{u\in N\left({v}_{m,j}^{q}\right)}d\left(u\right)=k+m+1$.

**Example**

**5.**

## 4. Concluding Remarks

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**1.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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${\mathit{e}}_{1}$ | ${\mathit{e}}_{2}$ | … | ${\mathit{e}}_{\mathit{m}}$ | |
---|---|---|---|---|

${v}_{1}$ | ${a}_{1,1}$ | ${a}_{1,2}$ | … | ${a}_{1,m}$ |

${v}_{2}$ | ${a}_{2,1}$ | ${a}_{2,2}$ | … | ${a}_{2,m}$ |

… | … | … | … | … |

${v}_{n}$ | ${a}_{n,1}$ | ${a}_{n,2}$ | … | ${a}_{n,m}$ |

${\mathit{v}}_{1}$ | ${\mathit{v}}_{2}$ | … | ${\mathit{v}}_{\mathit{n}}$ | |
---|---|---|---|---|

${v}_{1}$ | ${a}_{1,1}$ | ${a}_{1,2}$ | … | ${a}_{1,n}$ |

${v}_{2}$ | ${a}_{2,1}$ | ${a}_{2,2}$ | … | ${a}_{2,n}$ |

… | … | … | … | … |

${v}_{n}$ | ${a}_{n,1}$ | ${a}_{n,2}$ | … | ${a}_{n,n}$ |

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Wu, S.; Wang, M. Open Support of Hypergraphs under Addition. *Symmetry* **2022**, *14*, 669.
https://doi.org/10.3390/sym14040669

**AMA Style**

Wu S, Wang M. Open Support of Hypergraphs under Addition. *Symmetry*. 2022; 14(4):669.
https://doi.org/10.3390/sym14040669

**Chicago/Turabian Style**

Wu, Shufei, and Mengyuan Wang. 2022. "Open Support of Hypergraphs under Addition" *Symmetry* 14, no. 4: 669.
https://doi.org/10.3390/sym14040669