# On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus

^{1}

^{2}

^{*}

## Abstract

**:**

_{ij}], where for i≠j, d

_{ij}is the Euclidean distance between the nuclei i and j. In this matrix d

_{ii}can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclidean graph of a carbon nanotorus. We prove that this group is a semidirect product of a dihedral group by a group of order 2.

## 1. Introduction

## 2. Main Results and Discussion

_{ij}], where for i ≠ j, d

_{ij}is the Euclidean distance between the nuclei i and j. In this matrix d

_{ii}can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for different nuclei. Notice that a Euclidean graph is a complete edge weighted graph. By symmetry we mean the automorphism group symmetry of Euclidean graph of molecule under consideration. Here, an automorphism of a Euclidean graph G is a permutation g of the vertex set of G with the property that for any vertices u and v, d(g(u),g(v)) = d(u,v), where d(-,-) is usual Euclidean meter. The set of all automorphisms of a graph G, with the operation of composition of permutations, is a permutation group on V(G), denoted Aut(G).

^{t}EP = E, where E is the adjacency matrix of the molecule under consideration and P varies on the set of all permutation matrices with the same dimension as E. He computed the Euclidean graphs and automorphism group for benzene, eclipsed and staggered forms of ethane and eclipsed and staggered forms of ferrocene. One of the present authors (ARA), in some research papers [22,23,24,25,26,27,28] continued the leading works of Balasubramanian in computing symmetry of molecules.

#### 2.1. Computational Details

_{σ}given by P

_{σ}= [x

_{ij}], x

_{ij}= 1 if i = σ(j) and 0 otherwise. It is easy to see that P

_{σ}P

_{τ}= P

_{στ}, for any two permutations σ and τ on n objects, and so the set of all n × n permutation matrices is a group isomorphic to the symmetric group

**S**

_{n}on n symbols. It is a well-known fact that a permutation σ of the vertices of a graph G belongs to its automorphism group if and only if it satisfies P

_{σ}

^{t}AP

_{σ}= A, where A is the adjacency matrix of G.

_{σ})

^{t}DP

_{σ}= D, where D is the adjacency matrix of the Euclidean graph G of the nanotorus under consideration. Suppose Aut(G) = {σ

_{1}, σ

_{2},…, σ

_{m}}. The matrix S

_{G}= [s

_{ij}], where s

_{ij}= σ

_{i}(j) is called a solution matrix for G. Clearly, for computing the automorphism group of G, it is enough to calculate a solution matrix for G.

**Theorem**

**1.**

_{ij}] and B = [b

_{ij}] are two matrices and P

_{σ}is a permutation matrix. If B = P

_{σ}A(P

_{σ})

^{t}, σ(i) = r and σ(j) = s, then a

_{rs}= b

_{ij}. In particular, if B = A and σ maps i

_{1}→ j

_{1}, i

_{2}→ j

_{2},…, i

_{t}→ j

_{t}. Then we have:

_{1}, O

_{2}, …, O

_{t}are orbits of the action of G on X then for every α ∈ G and every positive integer i, 1 ≤ i ≤ t, α(O

_{i}) ∈ { O

_{1}, O

_{2}, …, O

_{t}O

_{1}, O

_{2}, …, O

_{t}}. We apply this fact and our theorem to prepare the following MATLAB program:

A MATLAB Program for Computing Solution Matrix |

function y=permute2(a) |

m=length(a); |

1:m; |

sort(a); |

r i=1:m |

x=[[]; |

for j=1:m |

if min(b(:,i)==b(:,j))==1 |

x=[x,j]; |

end |

end |

p(i,1:length(x))=x; |

end |

for i=1:m-2 |

for j=i+1:m |

if max(p(i,:)==j)==1 |

tt=0; |

s=[1:i-1 j]; |

for r=i+1:m |

n=size(s); |

w=[[]; |

for t=1:n(1) |

v=p(r,:); |

v(v==0)=[[]; |

k1=1:m;k1(v)=[[]; |

k=1:m; |

k([s(t,:) k1])=[[]; |

for f=k |

if min(a([s(t,:) f],[s(t,:) f])==a(1:r,1:r))==1 |

w=[w;s(t,:) f]; |

if r==m |

tt=1; |

break |

end |

end |

if tt==1 |

break |

end |

end |

if tt==1 |

break |

end |

end |

s=w; |

end |

if length(s)>1 |

y=[y; s(1,:)]; |

end |

end |

end |

end |

B:=[]; |

N:=Size(A); |

for i in [1,2..N] do |

d:=PermListList(A[1],A[i]); |

Add(B,d); |

od; |

G:=AsGroup(B); |

GeneratorsOfGroup(G); |

#### 2.2. Theoretical Results

^{−1}is still in N. Normal subgroups are important because they can be used to construct quotient groups from a given group. A semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal. Let G be a group, N a normal subgroup of G and H a subgroup of G. We say that G is a semidirect product of N and H, or that G splits over N, if every element of G can be written in one and only one way as a product of an element of N and an element of H.

_{n}) is the subgroup of S

_{n}generated by the permutations a = (1,2,…,n) and b =(2,n)(3,n-1)(4,n-2)…(n/2,n/2+2) or (2,n)(3,n-1)(4,n-2)…((n+1)/2,(n+3)/2), when n is even or odd, respectively. It is easy to see that D

_{n}is non-abelian, for n ≥ 3, and <a> is a normal subgroup of D

_{n}. It is possible to prove that D

_{n}is actually isomorphic to the group of symmetries of a regular polygon with n-sides. To explain, we consider a regular n-sided polygon P and label it clockwise by numbers 1, 2, …, n. Define the product operation of composition, as follows: for two symmetries f and g, the product fg means “first do f, then do g”. P has exactly n rotational symmetries: these are a, a

^{2}(= aa), …, a

^{n-1}and e = a

^{n}, which leaves the polygon fixed. Here a is a function such that a(1) = 2, a(2) = 3, …, a(n-1) = n and a(n) = 1. On the other hand, a

^{k}is rotation about the center of A through an angle 2πk/n. There are also n reflection symmetries: these are reflections in the n lines passing through the center of P and a corner or the midpoint of a side of the polygon. Suppose b is the reflection in the line through 1 and the center of P. Then b =(2,n)(3,n-1)(4,n-2)…(n/2,n/2+2) or (2,n)(3,n-1)(4,n-2)…((n+1)/2,(n+3)/2), when n is even or odd, respectively. Clearly, n reflections of P are b, ab, a

^{2}b, …, a

^{n-1}b. Thus all elements of the symmetry group of a polygon P constitute the same group as dihedral group D

_{n}.

**Theorem**

**2.**

_{p/2}and a plane symmetry group isomorphic to Z

_{2}, the cyclic group of order 2.

## 3. Conclusion

## Acknowledgements

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**MDPI and ACS Style**

Yavari, M.; Ashrafi, A.R.
On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus. *Symmetry* **2009**, *1*, 145-152.
https://doi.org/10.3390/sym1020145

**AMA Style**

Yavari M, Ashrafi AR.
On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus. *Symmetry*. 2009; 1(2):145-152.
https://doi.org/10.3390/sym1020145

**Chicago/Turabian Style**

Yavari, Morteza, and Ali Reza Ashrafi.
2009. "On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus" *Symmetry* 1, no. 2: 145-152.
https://doi.org/10.3390/sym1020145