We have considered possible ways of forming the simplest tetrahedral fullerenes, namely elementary tetrahedron C4, truncated tetrahedron C12, half-truncated cube C16, fullerenes C28 and C36. By analogy with ionic crystals, we introduced "mathematical" compounds, which form a topological cube of two tetrahedra inserted into each other, and construct graphs for them. Combined with the graph analysis, this approach allows obtain a clear knowledge of the tetrahedral fullerene structure. We extended our model to other tetrahedral fullerenes, in particular, tetrahedral fullerenes C64 and C76.