1. Introduction
The interest in nanotori started soon after the experimental discovery of the carbon nanotubes [
1]. A nanotube is a molecule with tubical shape comprised only by carbon atoms. The atoms in nanotubes are arranged in form of hexagons, and each atom is bonded with exactly three other carbon atoms. The discovery of nanotubes lead the researchers to believe that carbon nanotorus molecules might exist as well [
2], i.e., carbon molecules obtained by gluing the two ends of the nanotube. Soon afterwards, an experimental evidence of such molecules appeared [
3,
4]. Further research showed that these carbon nanotorus molecules have wide spectrum of properties. Certain species of carbon nanotori exhibit unusual magnetic properties, including persistent magnetic moments at nearly zero flux and colossal paramagnetic moments, and diverse variety of electric properties: some nanotori are conductors, while others are semiconducting or insulators [
5,
6,
7,
8,
9]. The properties of carbon nanotori are strongly related to their geometrical parameters, temperature, and the parameters of the nanotube used for their production [
10].
In chemistry, it is a common practice to represent molecules as graphs; each atom is a vertex and the bonds in the molecule are presented as edges in the graph. Nanotubical graphs, or just nanotubes, are graph representations of nanotubical molecules. Therefore, nanotubes are 3-connected, infinite, cubic planar graphs, and represented in a space they have tubical shape. A nanotube is obtained from a planar hexagonal grid by identifying objects (vertices, edges, faces) lying on two parallel lines, i.e., a hexagonal grid is wrapped into a tube. The wrapping of the hexagonal grid into a tube is determined by a pair of integers
such that
and
. For more details, see [
11,
12,
13].
Nanotorus is a simple 3-regular graph embedded on a torus with hexagonal faces only. The structure of nanotori is described [
8,
14,
15] by ordered pairs of integers
and
such that
,
, and
for all
. It can be considered so that the first pair defines the wrapping of the hexagonal grid into a tube, while the second pair defines the transformation of the (infinite) nanotube into a torus. There is a certain correspondence between these two vectors and some physical properties of the nanotorus. More precisely, these two vectors characterize the carbon nanotorus as a conductor, semiconductor, or insulator [
16]. Several papers considered the symmetry group of nanotubes [
17] and nanotori [
14,
18,
19] since the symmetry group brings the information about important properties of the band structure (electronic, phonon, etc.).
Similarly, graphs are used to represent network topology. Here, multiprocessors are presented as graphs with vertices representing processors and edges representing links between them. The network topology is crucial for interconnection network since it determines the performance of the network. Meshes and tori are among the most frequent multiprocessor networks available on the market [
20]. Nanotori are good alternative to torus interconnection networks in parallel and distributed applications [
21,
22,
23,
24]. Stojmenović [
23] introduced three different nanotori by adding edges on honeycomb meshes (honeycomb rectangular torus, honeycomb rhombic torus, and honeycomb hexagonal torus). This concept was generalized later in [
21], where a nanotorus was defined with three parameters. Generally, this definition is acquired in computer science, unlike the four parameters definition used in physics and chemistry.
These two approaches of constructing nanotori are inspired by real problems, while the oldest one has purely mathematical motivation. The oldest method for constructing nanotori was suggested by Altshuler [
25,
26] back in 1972/73. He considered maps on a torus, i.e., a cellular decomposition of a torus induced by a graph. A map is called regular map of type
if each cell/face is of size
and each vertex has degree
. It is obvious that the dual of a regular map of type
is a regular map of type
. The Handshake lemma and Euler’s equation for torus imply that the regular map on torus must be of type
,
or
. The regular maps of type
are nowadays known as nanotori, and they are in the focus of this article. In [
25,
26], there is a construction of all non-isomorphic nanotori on a given number of vertices, using three parameters,
,
. For each nanotorus, these three parameters are not uniquely determined, i.e., there are at most six different triplets of integers
constructing each nanotorus. The Altshuler construction was studied later in [
27,
28], in a slightly different notion. In [
27], the authors classify all regular maps on torus of type
or
. They give an explicit formula for the number of combinatorial types of these regular maps with
n vertices. These formulas are obtained in terms of arithmetic functions in elementary number theory. Similar study of this problem is given in [
28] as well.
The Altshuler construction differs from the four parameters construction (two parameters for each vector) given in more recent papers [
8,
15]. In [
25,
26], Altshuler showed that each nanotorus can be described by three parameters. This was not established for the other two constructions so far. In this paper, we show that these three approaches (the one with four parameters used in physics and chemistry, the three parameters construction used in computer science, known as the generalized honeycomb tori, and the three parameters construction by Altshuler) are equivalent, and as a consequence we obtain that that the first two approaches also construct all non-ispomorphic nanotori. Hence, each nanotorus can be determined by a quadruple or a triple (as generalized honeycomb torus or with respect to Altshuler) of integers. Even more, there are finitely many triples (as generalized honeycomb torus or with respect to Altshuler), but infinitely many quadruples of integers that describe the same nanotorus. In this article, we present explicit formulas that transform parameters of one construction into parameters of the other two constructions of nanotori.
4. Isomorphisms of Nanotori through Construction M
Isomorphism of maps
and
was described already in [
26]. In [
27], the authors study the number of nonisomorphic nanotori
with a given number of vertices. In a later paper [
28], authors studied the isomorphism between regular and chiral nanotori. Here, we present a different approach and we consider isomorphism of maps
and
. As a consequence, we show that there are infinitely many quadruples of integers that construct the nanotori isomorphic to
.
Let
be a nanotorus. As explained above, it is constructed from the infinite tiling of a plane by hexagons by cutting out the characteristic parallelogram
with vertices
,
,
, and
, and gluing properly opposite sides of
. There is an alternative description of the construction of
. Take again the infinite tiling of a plane by hexagons
with the standard basis
and
as described above (see
Figure 1). For every point
,
, identify
T with all points
, where
. Obviously, the resulting map is
. Observe that this construction immediately implies that for every pair of hexagons
A and
B in
, there is an automorphism of
mapping
A to
B. In other words, the dual of
is vertex-transitive.
Now take the infinite tiling of a plane by hexagons
and consider the set of points
Then
is precisely the set of points which are identified with
in the construction of
. We call it the mesh of points of
(see
Figure 6).
Let us choose a different pair of linearly independent vectors and . If , then and are obtained by identifying the same sets of points, so they are isomorphic. Obviously, and are isomorphic also if there exists an automorphism of which maps the set of points to . It is interesting that the opposite implication is also true. We have the following.
Proposition 12. Let and be two pairs of linearly independent vectors. Then and are isomorphic if and only if there is an automorphism of fixing and mapping to .
We remark that Proposition 12 is a direct consequence of the fact that a plane regularly covers a torus. Nevertheless, our proof is elementary and shows the importance of normal cycles, see also the examples below the proof.
Proof. Since the sufficiency is trivial, it suffices to prove that if and are isomorphic nanotori, then there is an automorphism of fixing and mapping to . To simplify the notation, denote by M and the maps and , respectively. In addition, denote by S and the sets and , respectively. Moreover, if is a point of which is a center of a hexagon, we denote by and the corresponding hexagons in M and , respectively.
As already mentioned, three normal cycles are passing through every hexagon in M. Consider the normal cycle created by hexagons and denote its length by n. Then, forms another normal cycle. Since every hexagon has exactly six neighbors in M, these two cycles are either identical or disjoint. Since M is a finite map, proceeding in this way we construct several disjoint cycles of hexagons, say r, while the -st cycle is identical with the first one. Consequently, is identified with for some m. We choose to be the smallest integer such that the previous statement is satisfied.
By the definition of normal cycle, we have
and no internal point of the line segment starting at
and terminating at
is in
S. Hence,
S contains only the points
on the line
. In addition,
if
. But
and also
for
. Extending this argumentation to the whole plane we get
and
M is isomorphic to
.
Figure 7 shows the nanotori
and
. Observe that these nanotori are isomorphic.
Now let be an isomorphism mapping M to . Since the duals of nanotori are vertex transitive, we may assume that . Let and be points of such that and . Then for the vectors and both connect centers of neighboring hexagons in ; hence, they have the same length as and , and is an equilateral triangle. Thus, there is an automorphism of fixing and mapping to and to .
Since
is an isomorphism, the chain of hexagons
is a normal cycle in
. For the very same reasons, the cycles
are different from
if
, while
is identical with
. Consequently,
is isomorphic to
for some
. (Observe that for this isomorphism one has to apply
.) However,
is a cycle of hexagons in
M which “turns left” by one hexagon at
and “turns right" by one hexagon at
. Hence,
must be a cycle of hexagons in
turning by one hexagon at
and oppositely turning by one hexagon at
since
is an isomorphism mapping
M to
. Thus,
and
Since
and
is an automorphism of
mapping
to
and
to
, we get the result. □
In the next two paragraphs we consider few examples. Characteristic parallelogram for
is given in
Figure 4a. Start at
and construct a normal cycle in direction of
. This normal cycle contains all hexagons of the nanotorus, hence
;
n equals the number of hexagons, i.e.,
. The center
is identified with
, i.e., with
,
(see
Figure 7). Thus,
is isomorphic to
,
. Choosing
, one obtains
and
. Recall that three normal cycles are passing through every hexagon, one in each direction
,
and
. Analogously, in the direction of
and
, we find isomorphic nanotori
and
,
, respectively.
In
Figure 4b, there is a characteristic parallelogram for
. Start at
and construct a normal cycle in direction of
. This normal cycle does not contain all hexagons of the nanotorus, there is also another normal cycle “parallel” to it. Hence,
and
.
Figure 8 shows that
. Thus,
is isomorphic to
,
. Plugging
, one obtains
. Transitivity implies that
and
are isomorphic nanotori.
In the sequel, we describe all automorphisms of fixing and then we find all pairs of linearly independent vectors and such that for the meshes of points .
The group of automorphisms of
fixing
is the dihedral group
. It consists of 6 rotations and 6 reflections. Let
be an anti-clockwise rotation by angle
around
. Observe that
maps
to
and
to
. Let
be a reflexion with axis
, where
. The elements of the group of automorphisms of
fixing
are
, where
is the identity. Let
be an arbitrary point of
. We have
Now we find all vectors and such that . We have the following statement.
Lemma 1. Let and be two pairs of linearly independent vectors. Then, if and only if there are such that , and .
Proof. Denote and . First we show the necessity of the conditions. Hence, suppose that . Since , all points of must be of the form for some . Hence, also and , the vectors determining the sides of the characteristic parallelogram for , must be of this form. So there are such that and there are such that .
By Proposition 1,
has exactly
hexagons while
has
of them. Thus,
.
On the other hand, suppose that there are
such that
,
and
. Obviously
since every point of
has the form
for some
, where
and
. It suffices to show that
. However, we have
If we denote the matrix by
, then (
1) states that
. Since
, we get
Since
, for
,
,
, and
we get
and
, which implies that
. □
Combining Proposition 12 and Lemma 1, we get the following statement.
Theorem 2. Let and be two linearly independent pairs of vectors. Then the nanotori and are isomorphic if and only if there are , and an automorphism of fixing and mapping to and to such that , and .
Or equivalently, by taking (*) into account, we have
Theorem 3. Let and be two pairs of linearly independent vectors. Then the nanotori and are isomorphic if and only if there are such that and at least one of the following holds
- 1:
and ;
- 2:
and ;
- 3:
and ;
- 4:
and ;
- 5:
and ;
- 6:
and .
Theorem 3 (
2) and (
3) imply that
,
, and
are isomorphic nanotori, see
Figure 9. Observe that the equation
has infinitely many solutions in
. This fact together with Theorems 2 and 3 imply the following.
Corollary 1. Each torus is isomorphic to infinitely many tori .
Redundancy of the Parameters of
Hence, the last corollary motivates us to look for quadruples having 0 as one of the four parameters. Indeed, we find n, r, and m, such that is isomorphic to . In what follows, we treat general nanotorus and naturally we assume that and are linearly independent vectors.
Corollary 2. Let x and y be integer solutions of the equation . Then, is isomorphic to , where , and .
Proof. Observe that b and d cannot be both equal to 0 since and are linearly independent. Without loss of generality we assume that . Choose and .
Let
and
be integer solutions of equation
. Then, for
we get
Since
, we get
Thus, is isomorphic to , by Theorem 3 (1). Since is isomorphic to for arbitrary by Theorem 3 (1), we get the result. □
By Theorem 3 (2), is isomorphic to , and by Theorem 3 (3), is isomorphic to also to . The parallelograms for and are obtained from the parallelogram for by and , respectively. Recall that three normal cycles are passing through every hexagon of . One normal cycle in is , i.e., moving in the direction of . The other two are moving in directions of and , respectively. The alternative way to obtain the other two normal cycles in is to move in direction of in and , respectively. Thus, we can generalize Corollary 2 to the following result.
Corollary 3. Let , , and . Denote for , and
, where are integer solutions of ;
, where are integer solutions of ;
, where are integer solutions of .
Then is isomorphic to for every i, .
As mentioned above, through every hexagon of three normal cycles are passing and, by Corollary 3, their lengths are , , and . Hence, this corollary gives us the possibility to represent with one side of the characteristic parallelogram being the vector , where n is the length of the longest normal cycle in the nanotorus, and there are infinitely many such representations In addition, the vector , the second side of the parallelogram, can be chosen such that .
Applying Corollary 3 to , we find that it is isomorphic to , , , , , , and infinitely many others. In this case and . If Corollary 3 is applied to , , , and respectively . Hence, the nanotori , , , , and infinitely many others are isomorphic to .
Proposition 6 and Corollary 3 imply the following.
Corollary 4. The nanotorus is isomorphic to where for are determined in Corollary 3.
Similarly, Proposition 9 and Corollary 3 imply the relation between the nanotorus and generalized honeycomb tori .
Corollary 5. The nanotorus is isomorphic to the generalized honeycomb torus where and for are determined in Corollary 3.
Applying Corollary 4 to
, we find that this nanotorus is isomorphic to
,
,
,
,
, and
. Analogously, by Corollary 5, one can find six different triplets
,
such that
is isomorphic to
. They are
,
,
,
,
, and
. It is worth mentioning that there are cases, such as
, that have only one representation with respect to Altshuler,
, or as generalized honeycomb torus,
. From these remarks it is obvious that a given nanotorus might have one or more (at most six) different Altshuler (resp. generalized honeycomb torus) representations. For more details, refer to the counting of the representations we refer to [
27,
28].