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Stone-Wales operators interchange four adjacent hexagons with two pentagon-heptagon 5|7 pairs that, graphically, may be iteratively propagated in the graphene layer, originating a new interesting structural defect called here Stone-Wales wave. By minimization, the Wiener index topological invariant evidences a marked anisotropy of the Stone-Wales defects that, topologically, are in fact preferably generated and propagated along the

On hexagonal systems like graphene layers, graphene nanoribbons (GNR’s) and carbon nanotubes (CNT’s), the isolated pentagon–heptagon _{q/r} (_{6/6} in graphene (_{5/6} in fullerenes (

Scope of this theoretical article is to illustrate the topological properties of SW rotations in hexagonal systems investigating, in particular, the family of isomeric SW transformations able to ^{2}-carbon systems recently appeared in literature (see [

Another interesting topological effect is also introduced, consisting in the diffusion of a 5|7 pair in the hexagonal network as a consequence of iterated SW rotations; this topology-based mechanism ^{2}-hybridized carbon rings, detected by scanning tunneling microscopy (STM) images, function as a quasi-one-dimensional metallic wire and may be the building blocks for new all-carbon electronic devices. This important experimental finding enforces meanwhile the theoretical role of the SW waves, that are in principle structurally simpler than the pentagons-octagons chain reported in [

The SW rotations applied in the present studies derive from the general (_{p/r} varying the internal connectivity of four generic carbon rings made of _{p/r} reversibly rotates the bond shared by the two rings

and the total number of carbon-carbon bonds

On the graphene ideal surface, made only of hexagonal faces, the SW_{6/6} rotation transforms four hexagons in two 5|7 adjacent pairs (_{n}_{60} fullerene, its 1812 isomers are grouped by the _{5/6} (_{60}-I_{h}) through one or more SW transformations [_{60} isomeric space starting from just one C_{60} isomer.

Theoretical investigations based on plane-wave density-functional methods [_{2v} symmetry; this barrier reaches 9 eV for hexagonal systems like nanotubes or large graphene portions. Using the extended Hückel method, enlarging the relaxation region around the SW defect, it can be found that the formation energy of a SW defect considerably decreases to 6.02 eV for a flat graphene fragment case. This result has been verified by using

Extended theoretical investigations [

Considering the above experimental and theoretical evidences of the structural stability of hexagonal systems with 5|7 defects, this theoretical note aims to investigate the topological, wave–like mechanisms leading the diffusion (or annihilation) of pentagon–heptagon pairs.

_{6/6} (_{6/7} turns the bond between the heptagon and the nearby shaded hexagon and inserts the 6|6 couple of shaded hexagons between the two original 5|7 pairs (this topological defect is also referenced in [_{6/7} will successively drift the 5|7 pairs in the lattice (along the dotted directions in (

SWW mechanism provides theoretical support to recent studies on graphenic structures. Some authors [

Theoretically, SW defects and isolated 5|7 pairs have been extensively investigated in [

Considering the very rich variety and complexity of all possible paths that SW waves may describe on the graphenic surface, involving a variable numbers of 5|7 pairs, this article just focuses on the topological properties exhibited by the linear propagation of the basic SW defect, the 5|7 double pair (_{p/r} rotations to just to the operators SW_{6/6} and SW_{6/7}. In spite of the apparent simplicity of our model,

It is really important to note that, more and more, various anisotropic effects are evidenced in literature [

Before modeling the SW wave propagation, it is worth introducing the graphic tool used to generate this kind of defects on the hexagonal structures. The effectiveness of such an algorithm derives from the choice to operate in the

The generation of the SW rotations is greatly facilitated by considering the dual representation of the graphene layer by assigning to each hexagonal face the corresponding 6-connected (starred) vertex. _{p/r}

On the graphene dual layer the SW_{6/6} rotation (_{6/7} type that rotate, in the dual space, the _{6/7} operator originates the topological SW wave in both lattice representations.

_{6/7} rearrangements, evidencing with the dashed arrows the increasing distance between the two 5|7 pairs of the original SW_{6/6} dislocation. At each step, the pentagon (shaded circle) and the heptagon (black circle) interchange their locations with those of two hexagons (white circles) producing the _{6/6} rotation of (_{6/7} rearrangements used to generate the dislocations in both spaces. A SW wave produces (

The dual space represents the natural arena for studying all sorts of SW flips, avoiding the graphical difficulties that one usually encounters in redistributing the carbon atoms and bonds in the direct lattice. One easily generates the _{p/r} to create novel classes of isomeric rearrangements, with rings made of various numbers of atoms, of fullerene (dimensionality

The some-how arbitrary definition of _{6/7} rotations.

Energetically, the situation is more articulated; as explained in the next Section, the lattice shows in fact anisotropic reactions to the propagation of the SW waves along different directions when a closed graphene fragment is considered.

In summary, on the armchair-oriented graphene (

_{6/7} rotates the

_{6/7} rotates the

Above diffusion processes apply to an isolated 5|7 dislocation monopole as well to the 5|7 double pair arising from a SW_{6/6} rearrangement. In the following we mainly study this latter case, focusing on the mechanisms leading to the creation of diagonal or vertical

It is worth noting that similar topological tools are used in other disciplines like in Biology where wave-like diffusion mechanisms model cells proliferation processes [

In concluding this Section we observe that from the pure topological point of view one may consider each lattice configuration illustrated in this work as the result of an _{n}

The nature of the SW waves is investigated here by means of graph-theoretical methods only, postponing the correlations to the energy of the system to future specific investigations on the subject. According to this approximated model, we assign to the Wiener index _{60} fullerene isomers, just the _{60}-I_{h}_{66} fullerene. Computationally, our assumption implies the topological minimum principle on

Current selection of the topological potential, privileges transformations of the graphene layer, increasing the system compactness. The same method has been recently used in simulating the growing steps of fullerene-like nanostructures on the graphene dual plane [_{66} fullerene [

For a chemical graph with _{ij}_{i}_{j}

On large structures, this distance-based invariant shows a remarkable polynomial behavior. For infinite one-dimensional graphs [^{3}, being that a particular case of the polynomial-like general formula ^{s} (with

For the dual graphene lattice in

where

Initially, the topological propagation of the SW waves has been simulated on the _{10}_{10}_{G10}

Let’s now generate and propagate the diagonal SW wave (in

The first diagonal SW_{6/6} flip produces the 5|7 double pair (

The subsequent SW_{6/7} rotation (_{10}

A different situation is encountered by simulating the vertical SW wave given in

It is worth noticing that both the heptagon-pentagon and hexagons-hexagons bonds generated by the vertical SW in the dual lattice form a π/3 angle with the armchair edge, see _{6/6} vertical rotation, the two new 5|7 pairs give _{G10}_{6/7} flip starts the vertical propagation (_{10}.

The influence of the size of the system on the reported topological anisotropy has been investigated by considering the closed dual graphene lattice _{25} with _{G25}_{η}_{=0} _{25} lattice suffuses that of G_{10} one in accordance with the idea that for infinite extended system it will resemble the potential well, within which the diagonal movement takes place. The _{5/2}_{G25}_{G10}

It is moreover very interesting to note that a similar ratio 98.03 characterizes also the previously reported _{10} an _{25} layers with one SW defect (_{5/2}

Our simulations on the _{25} lattice confirm that the diagonal propagation on large distances of the 5|7 pair is still favored (_{10} layer (_{25} as the tendency of the system to recover, for large

The anisotropy of the SW waves constitutes an important effect induced by the topological potential

The main achievements of the present results are summarized as follows:

The tendency of the 5|7 defects [_{6/6} flip) two 5|7 pairs and then separates them via consecutive SW_{6/7} rotations, creating an extended

In the graphene layer SW_{6/7} flips are also able to transport isolated 5|7 dislocation monopoles, by exchanging the heptagon and pentagon positions with those of two nearby hexagons; this drifting mechanism may also annihilate or modify the 5|7 pair in colliding with other structural defects (grain boundaries, other 5|7 pairs, generic

The preference for diagonal SW waves perfectly matches results in literature [

A further confirmation of the existence of a preferred direction for the diffusion of the SW pairs comes from the distinct pentagon-pentagon bond energy predicted in [_{25} lattice or the 6344 atoms square graphite sheets previously modeled in literature using molecular mechanics tools [

These studies reflect the relevance of the anisotropic behavior of SW defects and of the topological SW waves mechanism both introduced in this article. Further investigations, both theoretical and experimental, are required to fully understand the capability of the suggested mechanisms in producing stable edge reconstructions in graphenic systems.

Finally, we notice that the present topological model is applicable to a wide class of chemical structures including systems with vacant atoms or other kinds of structural defects or to describe the evolution, driven by the topological potential _{n}

The proposed topological potential

The dual representation of the systems makes the generation and the characterization of SW waves remarkably easy, allowing the fast iteration of arbitrary sequences of generic SW_{q}_{|}_{r}

An open question remains regarding the infinite iteration of steps of SW wave propagation that should stabilize the net to a given energetic value, with observable character; this may be treated through combining the present topological approach with the path-integral or propagator information [

OO and MVP are grateful Hagen Kleinert and Axel Pelster form hospitality at Free University Berlin in the summer of 2011 where the paper was completed; MVP thanks Romanian CNCS-UEFISCDI (former CNCSIS-UEFISCSU) project TE16/2010-2011 within the PN II-RU-TE-2009-1 framework for supporting the present work and the German Academic Exchange Service (Deutscher Akademischer Austausch Dienst) for the fellowship DAAD/A/11/05356/322 allowing this paper being developed at Free University of Berlin.

_{60}and some related species

_{60}fullerenes

_{66}fullerene

(_{q/r} changes a group of four proximal faces with _{6/6} reversibly flips four hexagons in a 5|7 double pair; (_{5/6} reversible flip on the fullerene surface.

(_{6/6} originates two 5|7 pairs (in gray); (_{6/7} splits the pairs by swapping one of them with two nearby hexagons (shaded). Dotted SW_{6/7} pushes the SW wave in the dashed direction; (

Dual representation of the graphene lattice obtained by replacing each hexagonal face by the central 6-connected graph node. Graphene plane is then equivalently tiled by hexagons (direct space) or by starred nodes (dual space). The x-periodic (y-periodic) direct graphene nanoribbon has the armchair (zig-zag) orientation. The framed unit cell has been used to build this 4 × 7 graphenic fragment.

(_{6/7} swaps the pair made by one pentagon (dashed circle) and one heptagon (black circle) with two connected hexagons (dotted circles); dotted arrow indicates the next available translation of the 5|7 pair; (

SW vertical wave in the dual graphene layer after four propagation steps (dashed arrows); SW_{6/7} swaps the pentagon (dashed circle) heptagon (black circle) pair with two hexagons (dotted circles); dotted arrow indicates the next possible translation of the 5|7 pair, induced by a SW_{6/7} rotation of the hexagon-heptagon diagonal dashed bond. The SW wave generates anti-diagonal hexagons-hexagons bonds with respect to the unrotated one.

(_{10} as a functions of the wave propagation steps _{25}, vertical SW waves (bottom) present a limited penetration (_{G}_{25}_{G}_{10} ≈ 97.77 between the _{G}_{25} _{G}_{10} _{5/2}