An Algorithm for Counting Spanning Trees in Labeled Molecular Graphs Homeomorphic to Cata-Condensed Systems

The algorithmic method of Gutman and Mallion (1993), for calculating the number of spanning trees in the (labeled) molecular graphs of cata-condensed systems containing rings of only one size, was subsequently generalized by John and Mallion (1996) to make it applicable to such systems comprising rings of more than one size; this latter algorithm is thus generally valid for enumerating the spanning trees in the molecular graphs of any cata-condensed system. This algorithmic philosophy is extended here in order to devise a procedure that is suitable for an even more general class of molecular graphsnamely, those homeomorphic to the molecular graphs of cata-condensed systems. An example of its use is illustrated by explicitly computing the numerical value for the complexity of a (hypothetical) pentacyclic network consisting of two four-membered rings, two five-membered rings, and a nine-membered ring, giving rise to a spanning-tree count entirely in accord with that predicted via the theorem of Gutman, Mallion, and Essam (1983)on the face of it, an apparently very different approach, based on the “generalized characteristic polynomial” of the inner dual of the graph in question.