**1. Introduction**

The carbon nanostructures are the materials whose molecular structure is derived from graphene—the hexagonal carbon plain lattice (**Figure 1**). Because of their electronic struc‐ ture, they are the promising materials for the construction of nanoscale devices (quantum wires, nonlinear electronic elements, transistors, molecular memory devices, or electron field emitters) and the inventions in the material science.

The planar geometry of the molecular surface is disrupted by the disclinations in the molecular structure that are most often presented by the pentagons and the heptagons in the hexagonal lattice. This change of the geometry is manifested by the positive or the negative curvature, respectively, that can be enlarged by the supply of higher number of the defects. In this way,

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by the supply of 1–5 pentagonal defects, we get conical structures with different values of the vortex angle (**Figure 2**).

One more defect can be added and a nanotube is created. This nanostructure can be considered closed as well as opened, i.e., without the cap that contains the pentagonal defects. The second case is more common (**Figure 3**, left part). The number of the defects can be increased up to 12, and in this way, a completely closed, spherical nanostructure arises (fullerene—**Figure 3**, middle part).

**Figure 1.** Hexagonal carbon plain lattice.

**Figure 2.** Conical nanostructures with different numbers of pentagonal defects in the tip.

**Figure 3.** Different kinds of graphene nanostructures: nanotube (left), fullerene (middle), wormhole (right).

Analogical manipulations with the graphene lattice can be made by the supply of the heptag‐ onal defects (**Figure 4**). For the case of 12 heptagonal defects, if they are placed appropriately, the wormhole structure is created (**Figure 3**, right part).

**Figure 4.** Hexagonal lattice disclinated by one heptagonal defect.

by the supply of 1–5 pentagonal defects, we get conical structures with different values of the

One more defect can be added and a nanotube is created. This nanostructure can be considered closed as well as opened, i.e., without the cap that contains the pentagonal defects. The second case is more common (**Figure 3**, left part). The number of the defects can be increased up to 12, and in this way, a completely closed, spherical nanostructure arises (fullerene—**Figure 3**,

vortex angle (**Figure 2**).

32 Recent Advances in Graphene Research

**Figure 1.** Hexagonal carbon plain lattice.

**Figure 2.** Conical nanostructures with different numbers of pentagonal defects in the tip.

the wormhole structure is created (**Figure 3**, right part).

**Figure 3.** Different kinds of graphene nanostructures: nanotube (left), fullerene (middle), wormhole (right).

Analogical manipulations with the graphene lattice can be made by the supply of the heptag‐ onal defects (**Figure 4**). For the case of 12 heptagonal defects, if they are placed appropriately,

middle part).

A lot of other variants of the graphitic nanostructures can be created using different combi‐ nations of the pentagonal and the heptagonal defects. Some of them are presented in **Figure 5**.

**Figure 5.** Less common forms of the graphene nanostructures: triple-walled nanotube (left), pillared graphene (mid‐ dle), nanotoroid (right).

We investigate the electronic properties of several kinds of the carbon nanostructures. After the explanation of the computational methods, we demonstrate how to utilize these methods for the purpose of the investigation of graphene and some simple forms of the nanostructures —different kinds of nanoribbons and their modifications. Then, we will concentrate on the calculation of the properties of more complicated forms—the graphitic nanocone [1–3] and the graphitic wormhole [4–6]. In the first case, we consider the influence of the additional effects like the spin-orbit coupling (SOC) and the boundary effects coming from the finite size and from the extreme curvature of the surface geometry in the tip. In the second case, we investigate the effects that arise in the place of the wormhole bridge. Here, two additional effects appear: first, the SOC arising in the connecting nanotube and second, the increase of the electron mass due to relativistic effects coming from the extreme curvature of the surface geometry. As a result, the chiral massive electrons should be observed.
