On the Complexity of Fullerenes and Nanotubes

Catalogue of Risks

“…Such measures of complexity should (Randić et al 2005): Consider a state with about 30,000 laws concerning taxes and 20,000 or 50,000 people working with these laws.…”

Information Entropy of Fullerenes

“…Fullerenes are perfect candidates for this approach as a variety of group symmetries of the fullerenes are combined with the strict rules of their structures. Indeed, fullerenes previously became objectives of the topological studies, − which allow explaining most of their chemical properties and especially their stabilities. Partly due to the topological considerations, the widely used “isolated pentagon rule” has been deduced …”

Combinatorial enumeration in chemistry

“…Nanotubes. Randic´et al 301 considered the problem of evaluating the complexities of 19 lower fullerenes having from 20 to 50 carbon atoms and of buckminsterfullerene C 60 . They selected this set of fullerenes because they had been already studied by Laidboeur et al 302 who depicted all these fullerenes by means of their Schlegel graphs and denoted their symmetries.…”

“…Randic´et al 301 studied the complexity of lower fullerenes by using the distance degree sequences DDSs for all symmetry non-equivalent vertices, the multiplicity m of the DDSs, which are defined as the cardinality of each equivalence class, the augmented degrees for all vertices of different equivalence class x (see text above), [292][293][294] which represent a measure of the local complexity, the average value of the augmented vertex-degrees x av and the sum of all augmented vertex-degrees as the overall complexity index Vx av . For illustration, we give below the values of these complexity measures for buckminsterfullerene (since we gave in Figure 20 Both x av and Vx av increase with the size of the fullerene.…”

“…In the same article, Randic´et al 301 also considered the complexity of singlewall carbon nanotubes. These nanotubes can be constructed by rolling a long strip of a planar graphene sheet.…”