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

Abstract

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 graphssnamely, 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)son the face of it, an apparently very different approach, based on the "generalized characteristic polynomial" of the inner dual of the graph in question.

Aromaticity and Ring Currents

“…The carbon atoms in this chain have been numbered consecutively. (We are here referring, in an informal and intuitive fashion, to the graph-theoretical aspects that underlie the ring-current concept; this is a topic that has claimed the present authors' attention over the years ,− but it is outside the scope of the present review; any interested reader requiring a more rigorous discussion is directed, for example, to refs and −276). Figure c shows, as dotted lines, those carbon−carbon σ-bonds that have to be added to the open-chain spanning tree of Figure b in order to reconstitute the carbon−carbon σ-bond network of the original (pyrene) molecule depicted in Figure a.…”

Complexity of Molecules

“…A lot of work on the enumeration of spanning trees for various classes of (molecular) graphs has been carried out in the past 25 years. [37][38][39][40][41][42][43][44][45] The interest in spanning trees is related to several chemical problems, such as the use of spanning trees in chemical kinetics, [33][34][35][36] in calculating the magnetic properties of conjugated systems by means of the ring-current model [46][47][48][49] in the framework of π-electron molecular orbital theory, 50 and in coding and enumerating polyhex hydrocarbons. 51 The number of spanning trees can be obtained in several ways.…”

“…Therefore, the number of spanning trees can only be used as a measure of complexity for monocyclic and polycyclic molecular graphs. A lot of work on the enumeration of spanning trees for various classes of (molecular) graphs has been carried out in the past 25 years. − The interest in spanning trees is related to several chemical problems, such as the use of spanning trees in chemical kinetics, − in calculating the magnetic properties of conjugated systems by means of the ring−current model − in the framework of π-electron molecular orbital theory, and in coding and enumerating polyhex hydrocarbons …”

Exhaustive Generation of Organic Isomers. 5. Unsaturated Optical and Geometrical Stereoisomers and a New CIP Subrule

“…Computational organic chemistry has greatly stimulated the interest in developing software able to recognize and manage 2D and 3D molecular structures to assist in solving problems of synthesis, structure elucidation, database managing , similarity, − drug design, , and others. Active research has been done in the organic isomer enumeration − and generation , field particularly in computer aided stereoisomer generation. , …”