On Three Constructions of Nanotori

: There are three different approaches for constructing nanotori in the literature: one with three parameters suggested by Altshuler, another with four parameters used mostly in chemistry and physics after the discovery of fullerene molecules, and one with three parameters used in interconnecting networks of computer science known under the name generalized honeycomb tori. Altshuler showed that his method gives all non-isomorphic nanotori, but this was not known for the other two constructions. Here, we show that these three approaches are equivalent and give explicit formulas that convert parameters of one construction into the parameters of the other two constructions. As a consequence, we obtain that the other two approaches also construct all nanotori. The four parameters construction is mainly used in chemistry and physics to describe carbon nanotori molecules. Some properties of the nanotori can be predicted from these four parameters. We characterize when two different quadruples deﬁne isomorphic nanotori. Even more, we give an explicit form of all isomorphic nanotori (deﬁned with four parameters). As a consequence, inﬁnitely many 4-tuples correspond to each nanotorus; this is due to redundancy of having an “extra” parameter, which is not a case with the other two constructions. This result signiﬁcantly narrows the realm of search of the molecule with desired properties. The equivalence of these three constructions can be used for evaluating different graph measures as topological indices, etc.


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 (k, ) such that 0 ≤ ≤ k and k > 0. 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 (k, ) and (m, n) such that k 2 + 2 = 0, m 2 + n 2 = 0, and (k, ) = α(m, n) for all α ∈ Z. 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 {p, q} if each cell/face is of size p ≥ 3 and each vertex has degree q ≥ 3. It is obvious that the dual of a regular map of type {p, q} is a regular map of type {q, p}. The Handshake lemma and Euler's equation for torus imply that the regular map on torus must be of type {3, 6}, {4, 4} or {6, 3}. The regular maps of type {6, 3} 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, m, n, r ∈ N, 0 ≤ r < m. For each nanotorus, these three parameters are not uniquely determined, i.e., there are at most six different triplets of integers (m, n, r) 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 {3, 6}, {4, 4} or {6, 3}. 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.

Three Constructions of Nanotori
A nanotorus is a map on a torus in which every vertex has degree 3 and all faces are hexagons. Here we present the three constructions of nanotori. We start with the four parameters construction [15]. Take an infinite tiling of a plane by hexagons. Then every vertex is incident with exactly 3 hexagons. We denote by A the center of one of these hexagons. Moreover, denote by − → v 1 and − → v 2 two vectors connecting A with centers A 1 and A 2 , respectively, of adjacent hexagons, such that the triangle AA 1 A 2 is equilateral and its vertices A, A 1 , A 2 are enumerated in anti-clockwise order. Then, form the standard basis of the hexagonal tiling (see Figure 1). Sometimes we where AA 2 A 3 is an equilateral triangle and its vertices A, A 2 , A 3 are in anti-clockwise order. By (x, y) we denote the vector x − → v 1 + y − → v 2 and by [x, y] we denote the point A + x − → v 1 + y − → v 2 . Take a quadruple of integers a, b, c and d, so that (a, b) and (c, d) are linearly independent, and denote B = [a, b], C = [a + c, b + d], and D = [c, d]. Observe that B, C, and D are centers of hexagons and ABCD is a parallelogram since A = [0, 0]. We denote this parallelogram by P. Now cut P from the plane and glue its opposite sides such that the points A, B, C, and D will be identified into a single point. The resulting surface will be orientable. In this way, we obtain a map on a torus, in which every face is hexagon and every vertex has degree 3. Hence, it is a nanotorus. We denote it by M(a, b, c, d) and we call P a characteristic parallelogram for M(a, b, c, d). A characteristic parallelogram P for M(3, 1, −1, 3) is given in Figure 2. Observe that from the point of view of incidence, M(a, b, c, d) is a unique map. However, it has infinitely many realizations in the space. It is because we can glue the opposite sides of P with a shift (see Section 3). Although these realizations are different in three dimensional space, they correspond to isomorphic maps. Therefore, they have the same graph-theoretic invariants.
Observe that M(a, b, c, d) has positive number of hexagons (and vertices) if and only if (a, b) and (c, d) are linearly independent. More insight into this gives the following lemma, which is also given in [15].
Since by (b) the area of P is |a − bc/d| · |d| · z, the area of the original parallelogram P is |ad − bc|z as well.
Thus, P has the area of |ad − bc| hexagons and so M(a, b, c, d) has exactly |ad − bc| hexagons. Since the number of hexagons times 6 equals twice the number of edges which is equal to three times the number of vertices, it follows that M(a, b, c, d) has 2|ad − bc| vertices.
Later in this article, we will show that there are infinitely many different quadruples of integers (a , b , c , d ) that define isomorphic nanotori to the nanotorus M(a, b, c, d).

Altshuler's Construction, M * (r, n, m * )
In [26], Altshuler gives a different construction from the construction given in Section 2.1. In fact, he gives the construction for duals of nanotori. That is, he constructed toroidal maps in which every face is a triangle and every vertex has degree 6. Similarly, he starts with the infinite hexagonal tiling of a plane and with A being the center of one hexagon. For positive integers r, n and an integer m * , such that 0 ≤ m * < n, he denotes This gives a parallelogram AB * C * D * . Now cut this parallelogram from the plane and glue the sides AD * and B * C * so that A is identified with B * and D * is identified with C * analogously as in the standard construction. The next gluing is made with a shift by m * − → v 1 . That is, AB * is glued with D * C * (the original edges of the parallelogram become circles after the first gluing), in the manner that A is identified with the point E * = D * + m * − → v 1 (therefore the requirement 0 ≤ m * < n). This procedure gives a nanotorus which we denote by M * (r, n, m * ). The construction of an M * (3, 4, 1) nanotorus is given in Figure 3. The next proposition (cf. [25]) follows trivially from the construction. In [26], Altshuler shows that every nanotorus is isomorphic to M * (r, n, m * ) for some triple of values r, n, and m * , where 0 ≤ m * < n, and he gives bounds for the number of nonisomorphic nanotori having v hexagons. For this he uses the notion of normal cycle. Let H 0 , H 1 , H 2 , . . . , H n−1 be a sequence of hexagons in a nanotorus such that H i and H i+1 share an edge e i for every 0 ≤ i < n (indexed modulo n). Now, we say that the sequence H 0 , H 1 , H 2 , . . . , H n−1 forms a normal cycle (of length n), if e i−1 and e i are antipodal edges on a hexagon, the H i for every 0 ≤ i < n (indexed modulo n). It is not obvious that a nanotorus has a normal cycle. Nevertheless, in an earlier work [25], Altshuler proved that there are three normal cycles passing thought every hexagon in a nanotorus. Cutting the nanotorus along a normal cycle gives the tube obtained after gluing AD * with B * C * as described above (observe that because of normal cycle one must glue AD * with B * C * without twist, since the normal cycle passes through both A and B * ). Examples of normal cycles are given in Figure 4. The number of normal cycles determines the upper bound of the number of isomorphic nanotori M * (r, n, m * ) with different parameters. As there are three normal cycles passing through every hexagon, the following holds.  Observation 1. Every nanotorus can be represented by at most six different triplets of integers by Altshuler's construction M * .

Generalized Honeycomb Torus Construction, GHT(a, b, c)
In some papers, just to mention few [21][22][23], nanotori are constructed in a different way using three parameters a, b, and c. Therein, nanotori are called (generalized) honeycomb tori, and are denoted by GHT(a, b, c). We adopt the following notation. For x ∈ Z and y ∈ N, we denote by (x) y the integer satisfying 0 ≤ (x) y < y and x ≡ (x) y (mod y). Let us assume that a and b are positive integers, and b is even. Let c be any integer such that 0 ≤ c < b and a − c is an even number. The generalized honeycomb torus GHT(a, b, c) is a structure with vertex set {(i, j) | 0 ≤ i < a, 0 ≤ j < b} such that (i, j) and (k, ) are adjacent if they satisfy one of the following conditions: The construction of generalized honeycomb torus GHT (3,8,7) is given in Figure 5. The vertices with the same labels are overlapping. (1, 7) (1, 0) (1, 1) (1, 2) (1, 0) (1, 7) (2, 6) (2, 7) Similarly as for the previous two constructions, the number of vertices and hexagons in a generalized honeycomb torus can be determined in the following way. Similarly to Altshuler's construction, there are finitely many generalized honeycomb tori isomorphic to each other. More precisely, by Observation 1 and by the results stated later (Propositions 4 and 5), there are at most six different triples of integers that define the same generalized honeycomb torus.
Note that the presented construction for some borderline values may not provide a cubic graph. For example, GHT(a, b, c): -for a = 1, b even, and c ∈ {1, b − 1}; or -for a ∈ N, b = 2 and c ∈ {0, 1}.
The same holds for the other two constructions. In what follows we will ignore such borderline cases.

Relation between the Constructions
Although we introduced three different constructions of nanotori, they all produce isomorphic structures. In what follows we prove this statement.

Relation between GHT and M *
First we consider the isomorphism between generalized honeycomb tori and nanotori defined according to Altshuler. (i * ) i = k and j = ± 1; and (ii * ) j = , and k = i − 1 and i + j is even.
Having the vectors − → v 1 and − → v 2 in the regular hexagonal tiling, as described in Section 2.1, we get that (i, j) ± − → v 1 = (i, j ± 2) and (i, j) ± − → v 2 = (i ± 1, j ± 1) (see Figure 5). The condition (i) identifies the vertices (i, j) and (i, j + b) in GHT(a, b, c) where b is even. Hence, the vertex (i, j) is identified with the vertex (i, j) + b 2 − → v 1 . This condition defines a nanotube. For j even, from the condition (iii) we have that the vertex (0, j), 0 ≤ j < b, is identified with the vertex (a, (j + c) b ). The vertex (a, (j + c) b ) in Z a × Z b can be expressed as This identification implies that the nanotorus GHT(a, b, c) is M * (a, b 2 , Directly from Proposition 4, we have the following.

Relation between Constructions M and M *
Next, we show that each nanotorus M(m, n, k, ) is isomorphic to nanotorus M * (r * , n * , m * ) for some r * , n * , m * ∈ N.
. Now cut the triangle AE * D * from the parallelogram P * and attach it by side AD * to B * C * . This gives a parallelogram P, which should be glued so that all its vertices are identified into a single point. Hence, this identification gives the map M(n, 0, m * − r, r).
If m * − r ≥ 0, we are done as 0 ≤ m * − r ≤ n − 1. Otherwise, denote by F * the vertex of P opposite to A so that P is AB * F * E * . Now cut the triangle AF * E * from P and attach it by side AE * to B * F * . This gives a parallelogram P and identifying its opposite sides we obtain M(n, 0, m * − r + n, r). Repeating this procedure several times, we obtain the result.

Relation between GHT and M
Next, we determine the relation between the generalized honeycomb torus and the nanotorus M, determined with four parameters. Similarly, like in the previous cases, from Propositions 4 and 6, we obtain the following propositions.
In the same manner, we find that the following holds.  M(a, b, c, d) is isomorphic to M * (r i , n i , (m i + r i ) n ) where n i , m i , r i for 1 ≤ i ≤ 3 are determined in Corollary 3. M(a, b, c, d) is isomorphic to the generalized honeycomb torus GHT(r i , 2n i , c i ) where c i = (2m i + r i ) 2n i and n i , m i , r i for 1 ≤ i ≤ 3 are determined in Corollary 3.

Proposition 11. The nanotorus
In order to prove Propositions 10 and 11, we describe a technique of finding whether two nanotori M (a, b, c, d) and M(a , b , c , d ) are isomorphic (Section 4).

Isomorphisms of Nanotori through Construction M
Isomorphism of maps M * (r 1 , n 1 , m * 1 ) and M * (r 2 , n 2 , m * 2 ) was described already in [26]. In [27], the authors study the number of nonisomorphic nanotori M * (r, n, m * ) 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 M(a 1 , b 1 , c 1 , d 1 ) and M(a 2 , b 2 , c 2 , d 2 ). As a consequence, we show that there are infinitely many quadruples of integers that construct the nanotori isomorphic to M(a, b, c, d).
Let M(a, b, c, d) be a nanotorus. As explained above, it is constructed from the infinite tiling of a plane by hexagons by cutting out the characteristic parallelogram P with vertices  S(a , b , c , d ), then M(a, b, c, d) and M(a , b , c , d ) are obtained by identifying the same sets of points, so they are isomorphic. Obviously, M(a, b, c, d) and M(a , b , c , d ) are isomorphic also if there exists an automorphism of H which maps the set of points S (a, b, c, d) to S(a , b , c , d ). It is interesting that the opposite implication is also true. We have the following.  S(a, b, c, d) to S(a , b , c , d ).
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. M(a, b, c, d) and M(a , b , c , d ) are isomorphic nanotori, then there is an automorphism of H fixing [0, 0] and mapping S(a, b, c, d) to S (a , b , c , d ). To simplify the notation, denote by M and M the maps M(a, b, c, d) and M(a , b , c , d ), respectively. In addition, denote by S and S the sets S(a, b, c, d) and S(a , b , c , d ) and M is isomorphic to M(n, 0, m, r). Figure 7 shows the nanotori M(3, 1, −1, 3) and M(10, 0, 3, 1). Observe that these nanotori are isomorphic.  For the very same reasons, the cycles

Proof. Since the sufficiency is trivial, it suffices to prove that if
and ψ is an automorphism of H mapping − → v 1 to − − → OV 1 and − → v 2 to − − → OV 2 , we get the result.
The group of automorphisms of H fixing [0, 0] is the dihedral group D 12 . It consists of 6 rotations and 6 reflections. Let ρ be an anti-clockwise rotation by angle 2π/6 = 60 • around [0, 0]. Observe that ρ Now we find all vectors (a , b ) and (c , d ) such that S(a, b, c, d) = S (a , b , c , d ). We have the following statement.
Or equivalently, by taking (*) into account, we have  M(a, b, c, d) and M(a , b , c , d ) are isomorphic if and only if there are x 1 , y 1 , x 2 , y 2 ∈ Z such that |x 1 y 2 − x 2 y 1 | = 1 and at least one of the following holds 1: Theorem 3 (2) and (3) imply that M(3, 1, −1, 3), M(−1, 4, −3, 2), and M(4, −1, 2, −3) are isomorphic nanotori, see Figure 9. Observe that the equation |x 1 y 2 − x 2 y 1 | = 1 has infinitely many solutions in Z. This fact together with Theorems 2 and 3 imply the following. Redundancy of the parameters of M(a, b, c, d) 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 M(a, b, c, d) is isomorphic to M(n, 0, m, r). In what follows, we treat general nanotorus M(a, b, c, d) and naturally we assume that (a, b) and (c, d) are linearly independent vectors. Similarly, Proposition 9 and Corollary 3 imply the relation between the nanotorus M(a, b, c, d) and generalized honeycomb tori GHT.

Corollary 5.
The nanotorus M(a, b, c, d) is isomorphic to the generalized honeycomb torus GHT(r i , 2n i , c i ) where c i = (2m i + r i ) 2n i and n i , m i , r i for 1 ≤ i ≤ 3 are determined in Corollary 3.

Concluding Remarks
We studied three, known in the literature, approaches for constructing nanotori. The first one was suggested by Altshuler. Later, inspired by by real life problems, two new and different approaches were introduced; one for describing nanotori molecules, and another one for describing interconnecting networks in computer science. In his work Altshuler showed that his method gives all non-isomorphic nanotori, but for the last two definitions there were no such results. Here we showed that the later two approaches also construct all nanotori. At the same time, we gave explicit formulas that convert parameters of one construction into the parameters of the other two constructions. The equivalence of these three constructions can be used for evaluating different graph measures as topological indices.
The four parameters construction is mainly used in chemistry and physics to describe carbon nanotori molecules. We show that there are infinitely many quadruples that determine the same nanotorus, and we characterize when two different quadruples define isomorphic nanotori. Even more, we gave an explicit form of all isomorphic nanotori (defined with four parameters). This result significantly narrows the realm of search of the molecule with desired properties.
Author Contributions: Conceptualization, R.Š., P.D. and M.K.; supervision, R.Š.; visualization, V.A.; writing-original draft, P.D. and M.K.; writing-review & editing, V.A. and R.Š. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.