taken into consideration in the case of endless layers.

Nanoribbons and nanotubes {#sec:Nanoribbons and nanotubes}
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The CC approach is especially applicable to impurities, as well as nanotubes and nanoribbons. In the latter two cases a group of atoms lying across the tubes or ribbons is a natural cluster, in which the number of atoms depends on the diameter of the tube respectively the width of the ribbon (Fig.\[Fig7\]). Let us now consider the armchair ribbon with $N=12$. As zigzag nanoribbons are not analogous to carbon nanotubes [@Wakabayashi2] the former will not be discussed in this work. The above mentioned cluster is shown in (Fig.\[Fig7b\]). This cluster differs from the cluster in Fig.\[Fig2b\] in a single, yet very significant aspect: the cluster in Fig.\[Fig7b\] has no closing bonds between atoms 1-6 and 7-12. This reflects the fact, that the latter atoms are situated on the edge and therefore are connected to the ribbon by means of two, not three bonds.

As one can see, the cluster in Fig.\[Fig7b\] belongs to a well known ladder type. The determinant $D_N$ of this cluster may be written in block form for any even $N$

$$\label{eq:17}
D_N = \left| \begin{matrix} L_{N_1} (\varepsilon)&I_{N_1}\\ I_{N_1}&L_{N_1} (\varepsilon) 