Designing mini-fullerenes and their relatives on graph basis

Different families of fullerenes taking as a basis their progenitors, mini-fullerenes, and graph theory were constructed. They are: a family of a bi-trefoil: C₁₄, C₁₈, C₂₄, C₃₀, C₃₆; a family of truncated bipyramids: C₁₄, C₁₈, C₂₄ C₃₀, C₃₆; a family of cupola half-fullerenes: C₁₀, C₁₂, C₁₆, C₂₀, C₂₄; and a family of 4-6-equator fullerenes: C₂₀, C₂₄, C₃₂, C₄₀, C₄₈. All the families contain five members and have a layer structure. By analogy with geography, we can distinguish two frigid zones composed of pentagons and a torrid zone composed of hexagons. The smallest fullerenes of each family are exceptions; its frigid zones are formed of squares. Another interesting feature is that the torrid zone in the family of 4-6-equator fullerenes: C₂₀, C₂₄, C₃₂, C₄₀, and C₄₈ is composed of square-hexagon pairs. We have also constructed a family of cupola half-fullerenes: C₁₀, C₁₂, C₁₆, C₂₀, and C₂₄ with the help of transition to modified graphs introduced in [3], where a cluster of three or four atoms being considered as a big point (vertex) contrary to a zero-size point (vertex) of a common graph.