Isomers of Polyenes Attached to Benzene

A polyene graph is a tree that can be embedded in a hexagonal lattice. Systems of polyene graphs attached to one hexagon are considered. Overlapping edges and/or vertices (geometrically nonplanar systems) are allowed. A complete mathematical solution is presented in terms of a generating function for the numbers of isomers of the systems in question. The corresponding geometrically planar systems, referred to as styrenoids, are enumerated by computer programming. Finally, in the Appendix, the generating function is given for the numbers of free polyene graphs

THEORY OF POLYPENTAGONS

Polypentagons are systems consisting of pentagons exclusively. Some of their topological properties are studied, including the relations between certain invariants. Complete mathematical solutions are reported for the numbers of polypentagons within certain classes: catacondensed systems (without internal vertices) and systems with one internal vertex and with two connected internal vertices. A complete account on proper polypentagons is given. These systems can, by definition, be embedded on a regular dodecahedron. It is found that exactly 39 such systems exist. Their chemical formulas (C(n)H(s)), forms, and symmetries are specified