# On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- 1.
- There exists an edge from each vertex${v}_{i}$to each vertex${w}_{j}$.
- 2.
- There is no edge from any vertex${v}_{i}$to any other vertex${v}_{k}$.
- 3.
- There is no edge from any vertex${w}_{j}$to any other vertex${w}_{l}$.

**Definition**

**3.**

**Example**

**1.**

## 2. Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

- $Y$ is a set of $uv$ elements;
- $H=\left\{{H}_{1},\dots ,{H}_{u}\right\}$ is a family of $u,$ $v$-sets or groups which form a partition of $Y$;
- $A$ is a family of $u$-sets or blocks of elements so that each $u$-set in $A$ intersects each group ${H}_{i}$ in exactly one element, and any pair of elements from different groups occurs together in exactly $w$ blocks in $A.$

**Theorem**

**1.**

**Example**

**3.**

**Theorem**

**2.**

- 1.
- $u-2$MOLS of order$v$;
- 2.
- a$T\left[u,1;v\right]$ transversal design;
- 3.
- an$OA\left[{v}^{2},u,v,2\right]$ orthogonal array.

**Example**

**4.**

**Definition**

**4.**

**Proposition**

**1.**

**Proof.**

**Example**

**5.**

- (I)
- The$n$mutually orthogonal$\left({K}_{1,1}{{\displaystyle \cup}}^{\text{\hspace{0.17em}}}\frac{n-1}{2}{K}_{1,2}\right)$-squares are${M}^{s}=\left({a}_{ij}^{s}\right),{a}_{ij}^{s}=\alpha ,i=\beta ,j=\alpha +s\beta +{\beta}^{2},n$is a prime$>2$and$s,\alpha ,\beta \in {\mathbb{Z}}_{n}$.
- (II)
- The$\left(n-1\right)$mutually orthogonal$\left(\left(n-2\right){K}_{1,1}{{\displaystyle \cup}}^{\text{\hspace{0.17em}}}{K}_{1,2}\right)$-squares are${M}^{s}=\left({a}_{ij}^{s}\right),$${a}_{ij}^{s}=\left(s+1\right)i+j-{c}_{i},s\in {\mathbb{Z}}_{n-1},nisaprime2,$and${c}_{i}=\{\begin{array}{lll}1\hfill & if\hfill & i=1,\hfill \\ 0\hfill & \hfill & otherwise.\hfill \end{array}$
- (III)
- If$n=9$, then$thethree$mutually$orthogonal{K}_{1,3}{{\displaystyle \cup}}^{\text{\hspace{0.17em}}}3{K}_{1,2}$-$squaresare$${M}^{s}=\left({a}_{ij}^{s}\right),s\in {\mathbb{Z}}_{3},{a}_{ij}^{s}=\beta ,i=\alpha ,j={\alpha}^{2}+s\alpha +\beta ,$and$\alpha ,\beta \in {\mathbb{Z}}_{9}$.
- (IV)
- I$f$$n=7$, then$thefour$mutually$orthogonal3{K}_{1,1}{{\displaystyle \cup}}^{\text{\hspace{0.17em}}}2{K}_{1,2}$-$squaresare{M}^{s}=\left({a}_{ij}^{s}\right),s\in {\mathbb{Z}}_{4},andi,j\in {\mathbb{Z}}_{7},$let$\beta \in {\mathbb{Z}}_{7}$,then${a}_{ij}^{s}=j,i=0,j\in {\mathbb{Z}}_{7},{a}_{ij}^{s}=\beta ,i=1,$$j=2+\beta +s,{a}_{ij}^{s}=\beta ,i=2,j=4+\beta +2s,{a}_{ij}^{s}=\beta ,i=3,j=6+\beta +3s,{a}_{ij}^{s}=\beta ,i=4,j=1+\beta +4s,{a}_{ij}^{s}=\beta ,i=5,j=4+\beta +5s,{a}_{ij}^{s}=\beta ,i=6,j=6+\beta +6s.$
- (V)
- The$n$mutually orthogonal${P}_{n+1}$-squares are${M}^{s}=\left({a}_{ij}^{s}\right),{a}_{ij}^{s}=\alpha ,i=\alpha +s\beta -{\beta}^{2},j=\alpha +\left(s+1\right)\beta -{\beta}^{2},\alpha ,\beta ,s\in {\mathbb{Z}}_{n}$where$n$is a prime greater than$2$; see [9].

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

^{T}, can be represented by the edge decomposition (as the graph squares), as shown in Figure 2. □

**Example**

**6.**

## 3. Recursive Constructions of the Graph-Orthogonal Arrays

**Definition**

**5.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proof.**

**Example**

**7.**

**Note:**The product$Y\times X$gives a new graph-orthogonal array different from the graph-orthogonal array constructed by the product $X\times Y.$Furthermore, we can generalize Proposition 3 by the following Proposition 4, which can be proven by the same technique followed in the proof of Proposition 3.

**Proposition**

**4.**

_{i}is a$\left({G}_{1}\otimes {G}_{2}\otimes \dots \otimes {G}_{h}\right)$-orthogonal array of order $m\times {\left({\prod}_{i=1}^{h}{n}_{i}\right)}^{2}$.

**Proof.**

## 4. Applications of the Graph-Orthogonal Arrays in the Design of Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**First edge decomposition of ${K}_{5,5}$ by ${P}_{4}{{\displaystyle \cup}}^{\text{\hspace{0.17em}}}2{P}_{2}$ corresponding to ${L}_{0}$.

**Figure 4.**Second edge decomposition of ${K}_{5,5}$ by ${P}_{4}{{\displaystyle \cup}}^{\text{\hspace{0.17em}}}2{P}_{2}$ corresponding to ${L}_{1}.$

**Figure 5.**Third edge decomposition of ${K}_{5,5}$ by ${P}_{4}{{\displaystyle \cup}}^{\text{\hspace{0.17em}}}2{P}_{2}$ corresponding to ${L}_{2}.$

${A}_{1}:{y}_{11}{y}_{21}{y}_{31}{y}_{41}$ | |

${A}_{2}:{y}_{11}{y}_{22}{y}_{32}{y}_{42}$ | |

${H}_{1}:{y}_{11}{y}_{12}{y}_{13}$ | ${A}_{3}:{y}_{11}{y}_{23}{y}_{33}{y}_{43}$ |

${H}_{2}:{y}_{21}{y}_{22}{y}_{23}$ | ${A}_{4}:{y}_{12}{y}_{21}{y}_{32}{y}_{43}$ |

${H}_{3}:{y}_{31}{y}_{32}{y}_{33}$ | ${A}_{5}:{y}_{12}{y}_{22}{y}_{33}{y}_{41}$ |

${H}_{4}:{y}_{41}{y}_{42}{y}_{43}$ | ${A}_{6}:{y}_{12}{y}_{23}{y}_{31}{y}_{42}$ |

${A}_{7}:{y}_{13}{y}_{21}{y}_{33}{y}_{42}$ | |

${A}_{8}:{y}_{13}{y}_{22}{y}_{31}{y}_{43}$ | |

${A}_{9}:{y}_{13}{y}_{23}{y}_{32}{y}_{41}$ |

Position | Coordinate Elements | Elements Determining Entries | Arrays |
---|---|---|---|

1, 1 | ${y}_{31},{y}_{41}\in {A}_{1}$ | with ${y}_{11},{y}_{21}$ | ${L}_{1}=\left[\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\\ 2& 3& 1\end{array}\right],{L}_{2}=\left[\begin{array}{ccc}1& 3& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]$ |

1, 2 | ${y}_{31},{y}_{42}\in {A}_{6}$ | with ${y}_{12},{y}_{23}$ | |

1, 3 | ${y}_{31},{y}_{43}\in {A}_{8}$ | with ${y}_{13},{y}_{22}$ | |

2, 1 | ${y}_{32},{y}_{41}\in {A}_{9}$ | with ${y}_{13},{y}_{23}$ | |

2, 2 | ${y}_{32},{y}_{42}\in {A}_{2}$ | with ${y}_{11},{y}_{22}$ | |

2, 3 | ${y}_{32},{y}_{43}\in {A}_{4}$ | with ${y}_{12},{y}_{21}$ | |

3, 1 | ${y}_{33},{y}_{41}\in {A}_{5}$ | with ${y}_{12},{y}_{22}$ | |

3, 2 | ${y}_{33},{y}_{42}\in {A}_{7}$ | with ${y}_{13},{y}_{21}$ | |

3, 3 | ${y}_{33},{y}_{43}\in {A}_{3}$ | with ${y}_{11},{y}_{23}$ |

Experimental Runs | Factor (Levels) | ||
---|---|---|---|

A | B | C | |

1 | 0 | 0 | 0 |

2 | 0 | 1 | 2 |

3 | 1 | 0 | 2 |

4 | 1 | 1 | 0 |

5 | 0 | 2 | 3 |

6 | 0 | 3 | 1 |

7 | 1 | 2 | 1 |

8 | 1 | 3 | 3 |

9 | 2 | 0 | 3 |

10 | 2 | 1 | 1 |

11 | 3 | 0 | 1 |

12 | 3 | 1 | 3 |

13 | 2 | 2 | 0 |

14 | 2 | 3 | 2 |

15 | 3 | 2 | 2 |

16 | 3 | 3 | 0 |

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**MDPI and ACS Style**

Higazy, M.; El-Mesady, A.; Mohamed, M.S.
On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares. *Symmetry* **2020**, *12*, 1895.
https://doi.org/10.3390/sym12111895

**AMA Style**

Higazy M, El-Mesady A, Mohamed MS.
On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares. *Symmetry*. 2020; 12(11):1895.
https://doi.org/10.3390/sym12111895

**Chicago/Turabian Style**

Higazy, M., A. El-Mesady, and M. S. Mohamed.
2020. "On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares" *Symmetry* 12, no. 11: 1895.
https://doi.org/10.3390/sym12111895