Compact Molecular Codes for Annulenes, Aza-annulenes, Annule- noannulenes, Aza-anmilenoannulenes, Cyclazines and Aza-cyclazines

The application of the recently introduced concept of compact molecular codes to monocyclic and bicycdic systems such as annulenes and annulenoannulenes is presented. In addition, this concept is reformulated to include heteroatoms. The compact molecular code in a novel formulation is applied to mono- cyclic, bicyclic and tricyclic heterosystems such as aza-annulenes, aza-annulenoannulenes, cyclazines and aza-cyclazines

A Theorem for Counting Spanning Trees in General Chemical Graphs and Its Particular Application to Toroidal Fullerenes

A theorem is stated that enables the number of spanning trees in any finite connected graph to be calculated from two determinants that are easily obtainable from its cycles → edges incidence-matrix. The 1983 theorem of Gutman, Mallion and Essam (GME), applicable only to planar graphs, arises as a special case of what we are calling the Cycle Theorem (CT). The determinants encountered in CT are the same size as those arising in GME when planar graphs are under consideration, but CT is applicable to non-planar graphs as well. CT thus extends the conceptual and computational advantages of GME to graphs of any genus. This is especially of value as toroidal polyhexes and other carbon-atom species embedded on the torus, as well as on other non-planar surfaces, are presently of increasing interest. The Cycle Theorem is applied to certain classic, and other, graphs – planar and non-planar – including a typical toroidal polyhex