$cos \frac{n_-\pi}{N_1+1} = \frac{1}{2}$ or $cos \frac{n_+ \pi}{N_1+1} = -\frac{1}{2}$, otherwise $\frac{n_- \pi}{N_1+1} = \frac{\pi}{3}$, $\frac{n_+ \pi}{N_1+1} = \frac{2\pi}{3}$; $n_+ = 2n$. Denoting $n_{-} = M$ one can write the both latter equations as

$$\label{eq:28}
N_1 = 3M - 1,$$

with M being an integer.

The formula is the same as has been obtained by means of the tight binding model in [@Fujita] and [@Nakada].

The wave functions of nanoribbons with metallic conductivity can be found by using the symmetry like it is done for the infinite layer in section\[sec:Wave functions\].

We consider as an example the case $N_1= 5$. The closed clusters for the description of the ribbon are shown in figures \[Fig8a\] and \[Fig8b\]. According to the graph theory terminology [@Trinajstic], these clusters are isomorphic to each other as well as to the ladder graph shown in Fig.\[Fig8c\]. The energy spectra of these graphs are therefore the same.

   

Likewise to the case of infinite layer one can also obtain the energy values of the cluster, which are corresponding to various symmetries. Results for $N_1=5$ are shown in Tab.\[table4\].

[&gt;m[1.2 cm]{}|&gt;m[1.2 cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}|&gt;m[1.2cm]{}]{}

&\
& [$-\sqrt{3} -1$ ]{}& & [-1]{} & & & & [$\sqrt{3} -1$ ]{}& & &\
& 