# A Structured Table of Graphs with Symmetries and Other Special Properties

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## Abstract

**:**

## 1. Introduction

## 2. Criteria and Optimal Graphs

#### 2.1. Criteria

#### 2.1.1. Diameter and MPL of a Regular Graph

#### 2.1.2. Bisection Bandwidth

#### 2.1.3. Automorphism Group Size

#### 2.2. Optimal Graphs

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Symbol | Description |
---|---|

${A}_{n}$ | Alternating group on a set of length n |

${C}_{n}$ | Cyclic group of order n |

${D}_{n}$ | Dihedral group of order $2n$ |

$\mathrm{GL}(n,p)$ | General linear group of degree n over finite field ${F}_{p}$ |

$\mathrm{PGL}(n,p)$ | Projective general linear group obtained from $GL(n,p)$ |

${S}_{n}$ | Symmetric group on a set of length n |

$\mathrm{Aut}\left(H\right)$ | Automorphism group of group H |

$\mathrm{Hol}\left(H\right)$ | Holomorph of group H |

$K\times H$ | Direct product of groups K and H |

${H}^{m}$ | Direct product of m copies of group H |

$K\u22caH$ | Semidirect product of groups K and H (In this manuscript, H acting faithfully on K) |

$K{\wr}_{n}H$ | Wreath product of groups K and H, H acting on ${K}^{n}$ (n is ommited when $H={S}_{n}$ or ${D}_{n}$) |

k | 3 | 4 | 5 | 6 | 7 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

N | |||||||||||

4 | 1.0000 | 1 | |||||||||

4 | 24 | ||||||||||

${S}_{4}$ | |||||||||||

5 | 1.0000 | 1 | |||||||||

6 | 120 | ||||||||||

${S}_{5}$ | |||||||||||

6 | 1.4000 | 1 | 1.2000 | 1 | 1.0000 | 1 | |||||

5 | 72 | 6 | 48 | 9 | 720 | ||||||

${S}_{3}\wr {S}_{2}$ | ${S}_{2}\wr {S}_{3}$ | ${S}_{6}$ | |||||||||

7 | 1.3333 | 1 | 1.0000 | 1 | |||||||

6 | 48 | 12 | 5040 | ||||||||

${D}_{4}\times {S}_{3}$ | ${S}_{7}$ | ||||||||||

8 | 1.5714 | 1 | 1.4286 | 1 | 1.2857 | 1 | 1.1429 | 1 | 1.0000 | 1 | |

4 | 16 | 8 | 1152 | 10 | 60 | 12 | 384 | 16 | 40320 | ||

${D}_{8}$ | ${S}_{4}\wr {S}_{2}$ | ${S}_{3}\times {D}_{5}$ | ${S}_{2}\wr {S}_{4}$ | ${S}_{8}$ | |||||||

9 | 1.5000 | 1 | 1.2500 | 1 | |||||||

8 | 72 | 14 | 1296 | ||||||||

${S}_{3}\wr {S}_{2}$ | ${S}_{3}\wr {S}_{3}$ | ||||||||||

10 | 1.6667 | 1 | 1.5556 | 1 | 1.4444 | 1 | 1.3333 | 1 | 1.2222 | 1 | |

5 | 120 | 8 | 320 | 13 | 28800 | 14 | 288 | 17 | 576 | ||

${S}_{5}$ | ${S}_{2}\wr {D}_{5}$ | ${S}_{5}\wr {S}_{2}$ | ${C}_{2}\times {S}_{3}\times {S}_{4}$ | ${D}_{4}\times ({S}_{3}\wr {S}_{2})$ | |||||||

11 | 1.6000 | 1 | 1.4000 | 1 | |||||||

8 | 22 | 16 | 5760 | ||||||||

${D}_{11}$ | ${C}_{2}\times {S}_{4}\times {S}_{5}$ | ||||||||||

12 | 1.9091 | 1 | 1.6364 | 1 | 1.5455 | 1 | 1.4545 | 1 | 1.3636 | 1 | |

6 | 18 | 10 | 48 | 12 | 576 | 18 | 1036800 | 20 | 4608 | ||

$(({C}_{2}\times {S}_{3})\wr {S}_{2})$ | |||||||||||

${D}_{9}$ | ${D}_{4}\times {S}_{3}$ | $\times {C}_{2}$ | ${S}_{6}\wr {S}_{2}$ | ${S}_{2}{\wr}_{6}({S}_{3}\wr {S}_{2})$ |

**Table 3.**The optimal graphs for $N=14,16,...,32;k=3$ (left) and for $N=13,14,...,21,32;k=4$ (right).

k | 3 | k | 4 | ||||
---|---|---|---|---|---|---|---|

N | N | ||||||

14 | 2.0769 | 1 | 13 | 1.6667 | 1 | ||

7 | 336 | 10 | 52 | ||||

$\mathrm{PGL}(2,7)$ | ${C}_{13}\u22ca{C}_{4}$ | ||||||

16 | 2.2000 | 2 | 14 | 1.6923 | 1 | ||

6 | 6 | 10 | 96 | ||||

${S}_{3}$ | ${{C}_{2}}^{2}\times {S}_{4}$ | ||||||

18 | 2.2941 | 1 | 15 | 1.7143 | 1 | ||

7 | 8 | 10 | 12 | ||||

${D}_{4}$ | ${D}_{6}$ | ||||||

20 | 2.3684 | 1 | 16 | 1.7500* | 1 | ||

8 | 20 | 12 | 32 | ||||

${D}_{10}$ | ${{C}_{2}}^{2}\wr {C}_{2}$ | ||||||

22 | 2.4805* | 1 | 17 | 1.8162* | 1 | ||

9 | 16 | 12 | 36 | ||||

${C}_{2}\times {D}_{4}$ | ${{S}_{3}}^{2}$ | ||||||

24 | 2.5652 | 1 | 18 | 1.8627* | 1 | ||

8 | 32 | 12 | 12 | ||||

$\mathrm{Hol}\left({C}_{8}\right)$ | ${D}_{6}$ | ||||||

26 | 2.6800 | 1 | 19 | 1.8889 | 1 | ||

9 | 52 | 12 | 24 | ||||

${C}_{13}\u22ca{C}_{4}$ | ${D}_{12}$ | ||||||

28 | 2.7778 | 1 | 20 | 1.9474 | 1 | ||

10 | 336 | 14 | 96 | ||||

$\mathrm{PGL}(2,7)$ | $\mathrm{GL}(2,3)\u22ca{C}_{2}$ | ||||||

30 | 2.8621 | 1 | 21 | 2.0000 | 2 | ||

9 | 1440 | 14 | 14 | ||||

$\mathrm{Aut}\left({S}_{6}\right)$ | ${D}_{7}$ | ||||||

32 | 2.9355 | 1 | 32 | 2.3548 | 3 | ||

10 | 12 | 18 | 3 | ||||

${A}_{4}$ | ${C}_{3}$ |

$(16,3)$ | $(21,4)$ | $(32,4)$ | $(32,4)$ | ||||
---|---|---|---|---|---|---|---|

2.2000 | 2 | 2.0000 | 2 | 2.3548 | 3 | 2.3548 | 3 |

6 | 6 | 14 | 14 | 18 | 3 | 18 | 3 |

${S}_{3}$ | ${D}_{7}$ | ${C}_{3}$ | ${C}_{3}$ |

$(\mathit{N},\mathit{k})$ | Semidirect Product Group | Finite Presentation |
---|---|---|

(26,3) | ${C}_{13}\u22ca{C}_{4}$ | $\langle a,b\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{a}^{13}={b}^{4}=1,ba{b}^{-1}={a}^{5}\rangle $ |

(13,4) | ||

(20,4) | $GL(2,3)\u22ca{C}_{2}$ | $\langle a,b,c,d,e\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}{a}^{4}={d}^{3}={e}^{2}=1,{b}^{2}={c}^{2}={a}^{2},$ |

$ab=ba,ac=ca,ad=da,cb{c}^{-1}={a}^{2}b,db{d}^{-1}={a}^{2}bc,$ | ||

$dc{d}^{-1}=b,eae={a}^{-1},ebe=bc,ece={a}^{2}c,ede={d}^{-1}\rangle $ |

k | N | Named Graphs |

4 | Tetrahedral graph; Complete; Distance-transitive; Strongly regular; Cayley; (3,3)-Cage; Circulant; Planar; Smallest cubic crossing number-0 | |

6 | Thomsen graph; Complete bipartite; Distance-transitive; Strongly regular; Cayley; (3,4)-Cage; Circulant; Smallest cubic crossing number-1 | |

8 | Wagner graph; Vertex-transitive; Cayley; Circulant; tripartite | |

10 | Petersen graph; Nonhamiltonian; Distance-transitive; Strongly regular; (3,5)-Cage; Smallest cubic crossing number-2; tripartite | |

3 | 14 | Heawood graph; Bipartite; Distance-transitive; Cayley; (3,6)-Cage; Smallest cubic crossing number-3 |

22 | Smallest cubic crossing number-7; tripartite | |

24 | McGee graph; (3,7)-Cage; Smallest cubic crossing number-8; tripartite | |

26 | Generalized Petersen-(13,5); Vertex-transitive; Smallest cubic crossing number-9; tripartite | |

28 | Coxeter graph; Nonhamiltonian; Distance-transitive; Smallest cubic crossing number-11; tripartite | |

30 | Levi graph; Bipartite; Distance-transitive; (3,8)-Cage; Smallest cubic crossing number-13 (to be proved) | |

5 | Pentatope graph; Complete; Distance-transitive; Strongly regular; Cayley; (4,3)-Cage; Circulant | |

6 | Octahedral graph; Complete tripartite; Distance-transitive; Strongly regular; Cayley; Planar; Circulant | |

8 | Complete bipartite; Distance-transitive; Strongly regular; Cayley; (4,4)-Cage; Circulant | |

4 | 9 | Generalized quadrangle-(2,1); (2, 3)-Hamming graph; (3, 3)-rook graph; 9-Paley graph; Distance-transitive; Strongly regular; Cayley; tripartite |

10 | Arc-transitive; Cayley; Circulant; tripartite | |

11 | 4-Andrásfai graph; Vertex-transitive; Cayley; Circulant; tripartite | |

12 | Vertex-transitive; Cayley; Circulant; tripartite | |

13 | 13-Cyclotomic graph; Arc-transitive; Cayley; Circulant; 4-partite | |

19 | Robertson graph; (4,5)-cage; tripartite | |

21 | Brinkmann graph; 4-partite (Table 4) | |

6 | Complete; Distance-transitive; Strongly regular; Cayley; (5,3)-Cage; Circulant | |

5 | 8 | (5,3)-Cone graph; 4-partite |

10 | Complete bipartite; Distance-transitive; Strongly regular; Cayley; (5,4)-Cage; Circulant | |

7 | Complete; Distance-transitive; Strongly regular; Cayley; (6,3)-Cage; Circulant | |

8 | 16-Cell; Complete 4-partite; Distance-transitive; Strongly regular; Cayley; Circulant | |

6 | 9 | Complete tripartite; Distance-transitive; Strongly regular; Cayley; Circulant |

10 | (6,4)-Cone graph; tripartite | |

12 | Complete bipartite; Distance-transitive; Strongly regular; Cayley; (6,4)-Cage; Circulant | |

7 | 8 | Complete; Distance-transitive; Strongly regular; Cayley; (7,3)-Cage; Circulant |

12 | Vertex-transitive; Cayley; Circulant; 4-partite |

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**MDPI and ACS Style**

Zhang, Y.; Huang, X.; Xu, Z.; Deng, Y.
A Structured Table of Graphs with Symmetries and Other Special Properties. *Symmetry* **2020**, *12*, 2.
https://doi.org/10.3390/sym12010002

**AMA Style**

Zhang Y, Huang X, Xu Z, Deng Y.
A Structured Table of Graphs with Symmetries and Other Special Properties. *Symmetry*. 2020; 12(1):2.
https://doi.org/10.3390/sym12010002

**Chicago/Turabian Style**

Zhang, Yidan, Xiaolong Huang, Zhipeng Xu, and Yuefan Deng.
2020. "A Structured Table of Graphs with Symmetries and Other Special Properties" *Symmetry* 12, no. 1: 2.
https://doi.org/10.3390/sym12010002