# A Survey on Symmetry Group of Polyhedral Graphs

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

Case 1. If $s=0$, then F is a fullerene and $|Aut(F)|$ divides 120.Cases 2–6. If s is either the number 1 or 2 or 3 or 4 or 5, then p is 10 or 8 or 6 or 4 or 2, respectively.Case 7. If $s=6$, then one can deduce that $p=0$ and thus F is a SPH-fullerene and $|Aut(F)|$ divides 24.

## 2. Main Results

**Proposition**

**1.**

**Theorem**

**1.**

**Theorem**

**2.**

- i.
- $(t,s,p)=(4,0,0)$:${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{D}_{8},{A}_{4},{S}_{4}.$
- ii.
- $(t,s,p)=(0,6,0):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{S}_{3},{D}_{8},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},\phantom{\rule{0ex}{0ex}}{\mathbb{Z}}_{2}\times {S}_{3},{D}_{12},{D}_{6},{\mathbb{Z}}_{2}\times {D}_{12},{\mathbb{Z}}_{2}\times {D}_{6},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}\times {D}_{6}.$
- iii.
- $(t,s,p)=(0,0,12):$${C}_{1},{\mathbb{Z}}_{2},{A}_{3},,{\mathbb{Z}}_{4},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{S}_{3},{S}_{6},{S}_{3},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{3},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{D}_{8},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{5},{D}_{12},{\mathbb{Z}}_{2}\times {S}_{3},\phantom{\rule{0ex}{0ex}}{A}_{4},{D}_{20},{\mathbb{Z}}_{2}{D}_{12},{D}_{24},{S}_{4},{A}_{4}\times {Z}_{2},{A}_{5},{\mathbb{Z}}_{2}\times {A}_{5}.$

**Theorem**

**3.**

- i.
- $(t,s,p)=(3,1,1)$:${C}_{1},{\mathbb{Z}}_{2}.$
- ii.
- $(t,s,p)=(3,0,3):$${C}_{1},{\mathbb{Z}}_{2},{A}_{3},{S}_{3},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{3}.$
- iii.
- $(t,s,p)=(2,3,0):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{S}_{3},{D}_{12}.$
- iv.
- $(t,s,p)=(2,2,2):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}.$
- v.
- $(t,s,p)=(2,1,4):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}.$
- vi.
- $(t,s,p)=(2,0,6):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{S}_{3},{\mathbb{Z}}_{2}\times {S}_{3},{D}_{12}.$
- vii.
- $(t,s,p)=(1,4,1):$${C}_{1},{\mathbb{Z}}_{2}.$
- viii.
- $(t,s,p)=(1,3,3):$${C}_{1},{\mathbb{Z}}_{2},{A}_{3},{S}_{3}.$
- ix.
- $(t,s,p)=(1,2,5):$${C}_{1},{\mathbb{Z}}_{2}.$
- x.
- $(t,s,p)=(1,1,7):$${C}_{1},{\mathbb{Z}}_{2}.$
- xi.
- $(t,s,p)=(1,0,9):$${C}_{1},{\mathbb{Z}}_{2},{A}_{3},{S}_{3}.$
- xii.
- $(t,s,p)=(0,5,2):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{5},{D}_{20}.$
- xiii.
- $(t,s,p)=(0,4,4):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{D}_{8},{\mathbb{Z}}_{4}.$
- xiv.
- $(t,s,p)=(0,3,6):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{A}_{3},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{3},{S}_{3},{D}_{12}.$
- xv.
- $(t,s,p)=(0,2,8):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{D}_{8},{\mathbb{Z}}_{2}\times {D}_{8},{D}_{16},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2},{\mathbb{Z}}_{4}.$
- xvi.
- $(t,s,p)=(0,1,10):$${C}_{1},{\mathbb{Z}}_{2},{\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}.$

## 3. Symmetry Group and Group Action

**Lemma**

**1.**

**(Cauchy–Frobenius Lemma)**Let G acts on set $V(F)$, then the number of orbits of G is

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Theorem**

**8.**

**Proof.**

**$|{G}_{{1}^{1}}|=|{({1}^{2})}^{{G}_{{1}^{1}}}|\times |{G}_{{1}^{1}}{}_{{}_{{1}^{2}}}|.$**It is easy to prove that $|{G}_{{1}^{1}}{}_{{}_{{1}^{2}}}|$ = 2, $|{({1}^{2})}^{{G}_{{1}^{1}}}|=1$ and $|{({1}^{1})}^{G}|=12$. Hence |G| = 24. On the other hand, $|\u2329\mathrm{a},\mathrm{b}\u232a|=24$ and this leads us to conclude that $G=\u2329\mathrm{a},\mathrm{b}\u232a\cong {D}_{24}$. Now suppose n is odd (see Figure 13), $fix(a)=\{{1}^{1},{1}^{2},{4}^{1},{10}^{2}\}$ and b = (1${}^{1}$, 1${}^{n+2}$, 2${}^{1}$, 2${}^{n+2}$, 2${}^{1}$, 2${}^{n+2}$, 2${}^{1}$, 2${}^{n+2}$, 2${}^{1}$, 2${}^{n+2}$, 2${}^{1}$, 2${}^{n+2}$) (1${}^{2}$, 2${}^{n+1}$, 4${}^{2}$, 5${}^{n+1}$, 7${}^{2}$, 8${}^{n+1}$, 10${}^{2}$, 11${}^{n+1}$, …, 16${}^{2}$, 17${}^{n+1}$) (2${}^{2}$, 3${}^{n+1}$, 5${}^{2}$, 6${}^{n+1}$, 8${}^{2}$, 9${}^{n+1}$, 11${}^{2}$, 12${}^{n+1}$, …, 17${}^{2}$, 18${}^{n+1}$) (3${}^{2}$, 4${}^{n+1}$, 6${}^{2}$, 7${}^{n+1}$, 9${}^{2}$, 10${}^{n+1}$, 12${}^{2}$, 13${}^{n+1}$, …, 18${}^{2}$, 1${}^{n+1}$) (1${}^{i}$, 3${}^{n-i+3}$, 5${}^{i}$, 7${}^{n-i+3}$, 9${}^{i}$, 11${}^{n-i+3}$, 13${}^{i}$, 15${}^{n-i+3}$, …., 21${}^{i}$, 23${}^{n-i+3}$) (2${}^{i}$, 4${}^{n-i+3}$, 6${}^{i}$, 8${}^{n-i+3}$, 10${}^{i}$, 12${}^{n-i+3}$, 14${}^{i}$, 16${}^{n-i+3}$, …, 22${}^{i}$, 24${}^{n-i+3}$) (3${}^{i}$, 5${}^{n-i+3}$, 7${}^{i}$, 9${}^{n-i+3}$, 11${}^{i}$, 13${}^{n-i+3}$, 15${}^{i}$, 17${}^{n-i+3}$, …, 23${}^{i}$, 1${}^{n-i+3}$) (4${}^{i}$, 6${}^{n-i+3}$, 8${}^{i}$, 10${}^{n-i+3}$, 12${}^{i}$, 14${}^{n-i+3}$, 16${}^{i}$, 18${}^{n-i+3}$, …, 24${}^{i}$, 2${}^{n-i+3}$) (1${}^{(n+3)/2}$, 3${}^{(n+3)/2}$, 5${}^{(n+3)/2}$, 7${}^{(n+3)/2}$, …, 21${}^{(n+3)/2}$, 22${}^{(n+3)/2}$) (2${}^{(n+3)/2}$, 4${}^{(n+3)/2}$, 6${}^{(n+3)/2}$, 8${}^{(n+3)/2}$,…, 22${}^{(n+3)/2}$, 24${}^{(n+3)/2}$), where (3≤i≤(n-3)/2). By similar argument $G=\u2329\mathrm{a},\mathrm{b}\u232a\cong {D}_{24}$. In general, we can conclude that the symmetry elements C${}_{24}$${}_{n}$ have the cycle types as given in Table 3. ☐

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

## 4. Leapfrog Operation

## 5. The Symmetry Group of Non-Classical Fullerenes

**Example**

**4.**

**Theorem**

**13.**

**Example**

**5.**

**Theorem**

**14.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Frucht, R. Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compos. Math.
**1938**, 6, 239–250. [Google Scholar] - Kroto, H.W.; Fichier, J.E.; Cox, D.E. The Fullerene; Pergamon Press: New York, NY, USA, 1993. [Google Scholar]
- Kroto, H.W.; Heath, J.R.; O’Brien, S.C.; Curl, R.F.; Smalley, R.E. C60: Buckminster fullerene. Nature
**1985**, 318, 162–163. [Google Scholar] [CrossRef] - Doslić, T. On lower bounds of number of perfect matchings in fullerene graphs. J. Math. Chem.
**1998**, 24, 359–364. [Google Scholar] [CrossRef] - Doslić, T. On some structural properties of fullerene graphs. J. Math. Chem.
**2002**, 31, 187–195. [Google Scholar] [CrossRef] - Doslić, T. Fullerene graphs with exponentially many perfect matchings. J. Math. Chem.
**2007**, 41, 183–192. [Google Scholar] [CrossRef] - Fowler, P.W.; Manolopoulos, D.E.; Redmond, D.B.; Ryan, R. Possible symmetries of fullerenes structures. Chem. Phys. Lett.
**1993**, 202, 371–378. [Google Scholar] [CrossRef] - Graver, J.E. Kekule structures and the face independence number of a fullerene. Eur. J. Combin.
**2007**, 28, 1115–1130. [Google Scholar] [CrossRef] [Green Version] - Graver, J.E. Encoding fullerenes and geodesic domes. SIAM. J. Discr. Math.
**2004**, 17, 596–614. [Google Scholar] [CrossRef] - Kutnar, K.; Marušič, D. On cyclic edge-connectivity of fullerenes. Discr. Appl. Math.
**2008**, 156, 1661–1669. [Google Scholar] [CrossRef] [Green Version] - Tang, A.; Li, Q.; Liu, C.; Li, J. Symmetrical clusters of carbon and boron. Chem. Phys. Lett.
**1993**, 201, 465–469. [Google Scholar] - Deza, A.; Deza, M.; Grishukhin, V. Fullerenes and coordination polyhedral graph versus half cube embeddings. Discret. Math.
**1998**, 192, 41–80. [Google Scholar] [CrossRef] [Green Version] - Deza, M.; Sikirić, M.D.; Fowler, P.W. The symmetries of cubic polyhedral graphs with face size no larger than 6. MATCH Commun. Math. Comput. Chem.
**2009**, 61, 589–602. [Google Scholar] - Deza, M.; Dutour, M. Zigzag structure of simple two-faced polyhedra. Comb. Probab. Comput.
**2005**, 14, 31–57. [Google Scholar] [CrossRef] - Fowler, P.W.; Cremona, J.E. Fullerenes containing fused triples of pentagonal rings. J. Chem. Soc. Faraday
**1997**, 93, 2255–2262. [Google Scholar] [CrossRef] - Dixon, J.D.; Mortimer, B. Permutation Groups; Springer-Verlag: New York, NY, USA, 1996. [Google Scholar]
- Krätschmer, W.; Lamb, L.D.; Fostiropoulos, K.; Huffman, D. Solid C
_{60}: A new form of carbon. Nature**1990**, 347, 354–358. [Google Scholar] [CrossRef] - Hirsh, A.; Brettreich, M. Fullerenes (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim). J. Am. Chem. Soc.
**2005**, 347, 409–411. [Google Scholar] - Moret, R. Structures, phase transitions and orientational properties of the C
_{60}monomer and polymers. Acta Cryst.**2005**, 61, 62–76. [Google Scholar] [CrossRef] - Randić, M. On the recognition of identical graphs representing molecular topology. Phys. Lett.
**1976**, 42, 283–287. [Google Scholar] [CrossRef] - Randić, M. On the recognition of identical graphs representing molecular topology. J. Chem. Phys.
**1974**, 60, 3920–3928. [Google Scholar] [CrossRef] - Balasubramanian, K. Graph theoretical perception of molecular symmetry. Chem. Phys. Lett.
**1995**, 232, 415–423. [Google Scholar] [CrossRef] - Balasubramanian, K. The symmetry groups of nonrigid molecules as generalized wreath products and their representations. Chem. Phys. Lett.
**1980**, 72, 665–677. [Google Scholar] [CrossRef] - Balasubramanian, K. Symmetry groups of chemical graphs. Int. J. Quantum Chem.
**1982**, 21, 411–418. [Google Scholar] [CrossRef] - Balasubramanian, K. Applications of combinatorics and graph theory to spectroscopy and quantum chemistry. Chem. Rev.
**1985**, 85, 599–618. [Google Scholar] [CrossRef] - Balasubramanian, K. Generating functions for the nuclear spin statistics of non-rigid molecules. J. Chem. Phys.
**1981**, 75, 4572–4585. [Google Scholar] [CrossRef] [Green Version] - Balasubramanian, K. Nonrigid Group Theory, Tunneling Splittings, and Nuclear Spin Statistics of Water Pentamer: (H
_{2}O_{5}). J. Phys. Chem.**2004**, 108, 5527–5536. [Google Scholar] [CrossRef] [Green Version] - Balasubramanian, K. Group theoretical analysis of vibrational modes and rovibronic levels of extended aromatic C48N12 azafullerene. Chem. Phys. Lett.
**2004**, 391, 64–68. [Google Scholar] [CrossRef] - Balasubramanian, K. Nuclear spin statistics of extended aromatic C
_{48}N_{12}azafullerene. Chem. Phys. Lett.**2004**, 391, 69–74. [Google Scholar] [CrossRef] [Green Version] - Babić, D.; Klein, D.J.; Sah, C.H. Symmetry of fullerenes. Chem. Phys. Lett.
**1993**, 211, 235–241. [Google Scholar] [CrossRef] - Boo, W.O.J. An introduction to fullerene structures. Geometry Symmetry Chem. Educ.
**1992**, 69, 605–609. [Google Scholar] - Fowler, P.W.; Quinn, C.M. σ,π and δ representations of the molecular point groups. Theoret. Chim. Acta
**1986**, 70, 333–350. [Google Scholar] [CrossRef] - Fowler, P.W.; Redmond, D.B. Symmetry aspects of bonding in carbon clusters: The leapfrog transformation. Theoret. Chim. Acta
**1992**, 83, 367–375. [Google Scholar] [CrossRef] - The GAP Team GAP, Groups, Algorithms and Programming, Lehrstuhl De für Mathematik; RWTH: Aachen, Germany, 1995.
- Kutnar, K.; Marušič, D.; Janezić, D. Fullerenes via their automorphism groups. MATCH Commun. Math. Comput. Chem.
**2010**, 63, 267–282. [Google Scholar] - Ghorbani, M.; Songhori, M. On the automorphism group of polyhedral graphs. App. Math. Comput.
**2016**, 282, 237–243. [Google Scholar] [CrossRef] - Ghorbani, M.; Songhori, M.; Ashrafi, A.R.; Graovać, A. Symmetry group of (3,6)-fullerenes. Fuller. Nanotub. Carbon Nanostruct.
**2015**, 23, 788–791. [Google Scholar] [CrossRef] - Ghorbani, M.; Songhori, M. Polyhedral graphs via their automorphism groups. App. Math. Comput.
**2018**, 321, 1–10. [Google Scholar] [CrossRef] - Ghorbani, M.; Ashrafi, A.R. Cycle Index of the Symmetry Group of Fullerenes C
_{24}and C_{150}. Asian J. Chem.**2007**, 19, 1109–1114. [Google Scholar] - Ghorbani, M. Enumeration of heterofullerenes: A survey. MATCH Commun. Math. Comput. Chem.
**2012**, 68, 381–414. [Google Scholar] - Ghorbani, M.; Dehmer, M.; Rajabi-Parsa, M.; Mowshowitz, A.; Emmert-Streib, F. On Properties of Distance-based Entropies on Fullerene Graphs. Entropy
**2019**, 21, 482. [Google Scholar] [CrossRef] [Green Version] - Ghorbani, M.; Dehmer, M.; Rajabi-Parsa, M.; Emmert-Streib, F.; Mowshowitz, A. Hosoya entropy of fullerene graphs. Appl. Math. Comput.
**2019**, 352, 88–98. [Google Scholar] [CrossRef] - Fowler, P.W. How unusual is C
_{60}, Magic numbers for carbon clusters. Chem. Phys. Lett.**1986**, 131, 444–450. [Google Scholar] [CrossRef] - Fowler, P.W.; Steer, J.I. The leapfrog principle- a rule for electron counts of carbon clusters. J. Chem. Soc. Chem. Commun.
**1987**, 1403–1405. [Google Scholar] [CrossRef]

Vertex | The Structure of Orbit Elements |
---|---|

1${}^{1}$ | ${1}^{1},{2}^{1},{3}^{1},{4}^{1},{5}^{1},{1}^{n+1},{2}^{n+1},{3}^{n+1},{4}^{n+1},{5}^{n+1}$ |

1${}^{2}$ | ${1}^{2},{3}^{2},{5}^{2},{7}^{2},{9}^{2},{2}^{n},{4}^{n},{6}^{n},{8}^{n},{10}^{n}$ |

2${}^{2}$ | ${2}^{2},{4}^{2},{6}^{2},{8}^{2},{10}^{2},{1}^{n},{3}^{n},{5}^{n},{7}^{n},{9}^{n}$ |

⋮ | ⋮ |

1${}^{n/2}$ | ${1}^{n/2},{3}^{n/2},{5}^{n/2},{7}^{n/2},{9}^{n/2},$ |

${2}^{(n+4)/2},{4}^{(n+4)/2},{6}^{(n+4)/2},{8}^{(n+4)/2},{10}^{(n+4)/2}$ | |

2${}^{n/2}$ | ${2}^{n/2},{4}^{n/2},{6}^{n/2},{8}^{n/2},{10}^{n/2},$ |

${1}^{(n+4)/2},{3}^{(n+4)/2},{5}^{(n+4)/2},{7}^{(n+4)/2},{9}^{(n+4)/2}$ | |

1${}^{(n+2)/2}$ | ${1}^{(n+2)/2},{2}^{(n+2)/2},{3}^{(n+2)/2},{4}^{(n+2)/2},{5}^{(n+2)/2},$ |

${6}^{(n+2)/2},{7}^{(n+2)/2},{8}^{(n+2)/2},{9}^{(n+2)/2},{10}^{(n+2)/2}$ |

Vertex | The Structure of Orbit Elements |
---|---|

1${}^{1}$ | ${1}^{1},{2}^{1},{3}^{1},{4}^{1},{5}^{1},{1}^{n+1},{2}^{n+1},{3}^{n+1},{4}^{n+1},{5}^{n+1}$ |

1${}^{2}$ | ${1}^{2},{3}^{2},{5}^{2},{7}^{2},{9}^{2},{1}^{n},{3}^{n},{5}^{n},{7}^{n},{9}^{n}$ |

2${}^{2}$ | ${2}^{2},{4}^{2},{6}^{2},{8}^{2},{10}^{2},{2}^{n},{4}^{n},{6}^{n},{8}^{n},{10}^{n}$ |

⋮ | ⋮ |

1${}^{(n+1)/2}$ | ${1}^{(n+1)/2},{3}^{(n+1)/2},{5}^{(n+1)/2},{7}^{(n+1)/2},{9}^{(n+1)/2},$ |

${1}^{(n+3)/2},{3}^{(n+3)/2},{5}^{(n+3)/2},{7}^{(n+3)/2},{9}^{(n+3)/2}$ | |

2${}^{(n+1)/2}$ | ${2}^{(n+1)/2},{4}^{(n+1)/2},{6}^{(n+1)/2},{8}^{(n+1)/2},{10}^{(n+1)/2},$ |

${2}^{(n+3)/2},{4}^{(n+3)/}$ |