#
Enumeration of Strongly Regular Graphs on up to 50 Vertices Having S_{3} as an Automorphism Group

## Abstract

**:**

## 1. Introduction

## 2. Orbit Matrices of Strongly Regular Graphs

**Definition**

**1.**

**Definition**

**2.**

## 3. The Method of Construction

- Construction of orbit matrices for the presumed automorphism group
- Construction of strongly regular graphs from the obtained orbit matrices (indexing of orbit matrices)

**Theorem**

**1.**

#### 3.1. Prototypes for a Row of a Column Orbit Matrix

#### 3.1.1. Prototypes for a Fixed Row

#### 3.1.2. Prototypes for a Non-Fixed Row

**Theorem**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**3.**

**Definition**

**5.**

**Lemma**

**1.**

## 4. SRGs with up to 50 Vertices Having ${S}_{3}$ as an Automorphism Group

#### 4.1. SRGs(37,18,8,9)

**Theorem**

**4.**

#### 4.2. SRGs(41,20,9,10)

**Theorem**

**5.**

#### 4.3. SRGs(45,22,10,11)

**Theorem**

**6.**

#### 4.4. SRGs(49,18,7,6)

**Theorem**

**7.**

#### 4.5. SRGs(49,24,11,12)

**Theorem**

**8.**

#### 4.6. SRGs(50,21,8,9)

**Theorem**

**9.**

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SRG | Strongly regular graph |

## References

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Distribution | #OM-${\mathit{S}}_{3}$ | #OM-${\mathit{Z}}_{3}$ | #SRGs |
---|---|---|---|

(1,0,0,6) | 3 | 6 | 0 |

(1,0,4,4) | 3 | 3 | 0 |

(1,0,8,2) | 3 | 3 | 0 |

Distribution | #OM-${\mathit{S}}_{3}$ | #OM-${\mathit{Z}}_{3}$ | #SRGs |
---|---|---|---|

(1,2,4,4) | 10 | 10 | 80 |

|Aut($\Gamma $)| | #SRGs |
---|---|

6 | 80 |

Distribution | #OM-${\mathit{S}}_{3}$ | #OM-${\mathit{Z}}_{3}$ | #SRGs |
---|---|---|---|

(1,4,4,4) | 7 | 7 | 288 |

|Aut($\Gamma $)| | #SRGs |
---|---|

6 | 288 |

Distribution | #OM-${\mathit{S}}_{3}$ | #OM-${\mathit{Z}}_{3}$ | #SRGs | Distribution | #OM-${\mathit{S}}_{3}$ | #OM-${\mathit{Z}}_{3}$ | #SRGs |
---|---|---|---|---|---|---|---|

(0,2,3,6) | 8 | 16 | 6 | (1,3,0,7) | 6 | 6 | 0 |

(0,2,5,5) | 4 | 0 | 0 | (1,3,2,6) | 10 | 2 | 0 |

(0,2,7,4) | 8 | 0 | 0 | (1,3,6,4) | 2 | 2 | 0 |

(1,0,0,8) | 2 | 15 | 2 | (1,6,0,6) | 1 | 0 | 0 |

(1,0,2,7) | 20 | 32 | 0 | (3,2,0,7) | 4 | 10 | 0 |

(1,0,8,4) | 26 | 24 | 12 | (3,2,6,4) | 6 | 16 | 0 |

(1,0,10,3) | 2 | 0 | 0 | (4,0,9,3) | 6 | 0 | 0 |

(1,0,12,2) | 16 | 0 | 0 | (5,1,0,7) | 2 | 4 | 0 |

(1,0,14,1) | 12 | 0 | 0 | (5,1,6,4) | 2 | 2 | 12 |

(7,0,0,7) | 2 | 2 | 40 |

|Aut($\Gamma $)| | #SRGs |
---|---|

6 | 42 |

18 | 22 |

24 | 4 |

126 | 4 |

Distribution | #OM-${\mathit{S}}_{3}$ | #OM-${\mathit{Z}}_{3}$ | #SRGs | Distribution | #OM-${\mathit{S}}_{3}$ | #OM-${\mathit{Z}}_{3}$ | #SRGs |
---|---|---|---|---|---|---|---|

(0,1,2,7) | 10 | 3 | 2 | (2,0,2,7) | 10 | 16 | 0 |

(0,1,4,6) | 10 | 4 | 6 | (2,0,8,4) | 22 | 24 | 12 |

(0,1,6,5) | 12 | 21 | 6 | (2,0,10,3) | 2 | 0 | 0 |

(0,1,8,4) | 8 | 8 | 1 | (2,0,12,2) | 27 | 0 | 0 |

(0,4,2,6) | 2 | 2 | 0 | (2,0,14,1) | 14 | 0 | 0 |

(0,4,4,5) | 4 | 3 | 16 | (2,3,0,7) | 2 | 3 | 0 |

(0,4,6,4) | 3 | 4 | 0 | (2,3,2,6) | 6 | 1 | 0 |

(0,4,8,2) | 4 | 6 | 0 | (2,3,6,4) | 2 | 4 | 0 |

(1,2,3,6) | 10 | 20 | 5 | (4,2,6,4) | 6 | 12 | 0 |

(1,2,5,5) | 2 | 0 | 0 | (5,0,9,3) | 2 | 0 | 0 |

(1,2,7,4) | 4 | 0 | 0 | (6,1,6,4) | 1 | 1 | 4 |

|Aut($\Gamma $)| | #SRGs |
---|---|

6 | 35 |

18 | 6 |

72 | 1 |

150 | 1 |

336 | 1 |

504 | 1 |

$(\mathit{v},\mathit{k},\mathit{\lambda},\mathit{\mu})$ | |Aut($\Gamma $)| | #SRGs |
---|---|---|

$(41,20,9,10)$ | 6 | 80 |

$(45,22,10,11)$ | 6 | 288 |

$(49,18,7,6)$ | 6 | 18 |

$(49,18,7,6)$ | 12 | 2 |

$(49,18,7,6)$ | 18 | 2 |

$(49,18,7,6)$ | 24 | 4 |

$(49,18,7,6)$ | 48 | 1 |

$(49,18,7,6)$ | 72 | 4 |

$(49,18,7,6)$ | 126 | 1 |

$(49,18,7,6)$ | 144 | 2 |

$(49,18,7,6)$ | 1008 | 1 |

$(49,18,7,6)$ | 1764 | 1 |

$(49,24,11,12)$ | 6 | 42 |

$(49,24,11,12)$ | 18 | 22 |

$(49,24,11,12)$ | 24 | 4 |

$(49,24,11,12)$ | 126 | 4 |

$(50,21,8,9)$ | 6 | 35 |

$(50,21,8,9)$ | 18 | 6 |

$(50,21,8,9)$ | 72 | 1 |

$(50,21,8,9)$ | 150 | 1 |

$(50,21,8,9)$ | 336 | 1 |

$(50,21,8,9)$ | 504 | 1 |

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**MDPI and ACS Style**

Maksimović, M.
Enumeration of Strongly Regular Graphs on up to 50 Vertices Having *S*_{3} as an Automorphism Group. *Symmetry* **2018**, *10*, 212.
https://doi.org/10.3390/sym10060212

**AMA Style**

Maksimović M.
Enumeration of Strongly Regular Graphs on up to 50 Vertices Having *S*_{3} as an Automorphism Group. *Symmetry*. 2018; 10(6):212.
https://doi.org/10.3390/sym10060212

**Chicago/Turabian Style**

Maksimović, Marija.
2018. "Enumeration of Strongly Regular Graphs on up to 50 Vertices Having *S*_{3} as an Automorphism Group" *Symmetry* 10, no. 6: 212.
https://doi.org/10.3390/sym10060212