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 graphsnamely, 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.