We have systematized possible ways of forming the isomers of mini-fullerenes, namely elementary fullerenes: tetrahedron C4, triangular prism C6, cube C8, pentagonal prism C10, hexagonal prism C12, as well as their derivatives, which were obtained by joining elementary fullerenes. Combined with the graph analysis, this approach allows obtain a clear knowledge of their structure. Among them there are barrel-shaped fullerenes: C12, C16, C20; tetrahedral ones C12 and C16; bi-shamrocks C14 and C18, bipyramids C14 and C18; regular and irregular dodecahedrons C20 as well as intermediate compounds. The three simplest elementary fullerenes, C4, C6, C8; have only electronic isomers and no space atomic isomers at all. After a cube, the next in size carbon fullerene C10 is a pentagonal prism. We have designed an isomer of it by fusion of a tetrahedron and a triangular prism. For the pentagonal prism shape fullerene the energy lies in the range from 974 to 2464 kJ/mol, for the hybrid of a tetrahedron and a triangular prism does in the range from 1396 to 2433 kJ/mol; it depends both on the number of single and double bonds as well as on their position in space. Fullerene of twelve carbon atoms C12 produces four isomers: a hexagonal prism, a barrel-shape fullerene, a truncated tetrahedron and a tetra-penta octahedron. They have different energies depending on the number of single and double bonds and their position in space. In a like manner other fullerenes studied, C14, C16, C18 and C20, have two or three isomers with different energies.