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Symmetries of Hexagonal Molecular Graphs on the Torus

Following the discovery and synthesis of spheroidal fullerenes, a natural question arises as to whether there exist torus-shaped graphite-like carbon structures which were given different names such as toroidal graphitoides or torusenes (see for instance Refs. 1–5). Indeed, they have been recently experimentally detected (see Ref. 4). Their graphs are cubic (trivalent), embedded onto torus. The geometry of these objects might afford new opportunities for holding reacting substrates in position. Unlike ordinary fullerenes that need the presence of twelve pentagonal faces, torusenes can be completely tessellated by hexagons.