formulated in sec.\[sec:Ham\] does not require these diagrams to be plane. Therefore, they also can be applied to nanotubes as well as spherical surfaces (fullerene). Stratified cluster can also be used. For example, the diagram in Fig.\[Fig2a\] can be rearranged to a hexahedron without breaking the bonds. The application for the space diagram is useful for studying many-layer graphene. In the next section we use a three-dimensional diagram to study bilayer graphene.

Bilayer graphene {#sec:Bilayer graphene}
================

For $N=12$, the 3D closed cluster corresponding to bilayer graphene is shown in Fig.\[Fig4\]. In addition to the closing bonds within the layers (energy $\gamma_0$), also the bonds describing the interactions between those layers (energy $\gamma_1$ in $\gamma_0$-units) are present in the cluster.

Note, that Fig.\[Fig4\] does not show the displacement of the upper layer towards the lower, since this Figure doesn´t reproduce a real atomic structure, but is solely a diagram which is used for building a Hamiltonian matrix.

![Cluster $N=12$ for bilayer graphene.[]{data-label="Fig4"}](./Fig4){width="5.5cm"}

The secular equation corresponding to Fig.\[Fig4\] has the form

$$\label{eq:8}
 D_{12}^{(2)} = \left|   
 \begin{array}  {rrrrrrrrrrrrr} 

     \varepsilon & 1 &   & 1  &   & 1 &   &   &   &  &  & \gamma_1\\
     1 & \varepsilon & 1 &  & 