Chemical applications of topology and group theory

Earlier approaches to the analysis of chemical dynamic systems using kinetic logic are refined to deal more effectively with systems having the two or more feedback circuits required for chaos. The essential kinetic features of such a system can be represented by a directed graph (called aninfluence diagram) in which the vertices represent the internal species and the directed edges represent kinetic relationships between the internal species. Influence diagrams characteristic of chaotic chemical systems have the following additional features: (1) They are connected; (2) Each vertex has at least one edge directed towards it and one edge directed away from it; (3) There is at least one vertex, called a turbulent vertex, with at least two edges directed towards it. From such an influence diagram a state transition diagram representing the qualitative dynamics of the system can be obtained using the following 4-step procedure: (1) A logical relationship is assigned at each turbulent vertex; (2) A local truth table is generated for each circuit in the influence diagram; (3) The local truth tables are combined to give a global truth table using the logical relationships at the turbulent vertices; (4) The global truth table is used to determine the corresponding state transition diagram using previously described methods. This refined procedure leads to a more restricted set of influence diagrams having the interlocking cycle flow topology required for chaos than the procedure described earlier. Systems with 3 internal species are examined in detail using the refined procedure. All systems with 3 dynamic variables shown in the simulation studies of Rössler to give chaotic dynamics correspond to influence diagrams which give inter-locking cycle (chaotic) flow topologies by the refined procedure. In addition, two models for the Belousov-Zhabotinskii reaction are examined using the refined procedure. The results are potentially informative concerning possible mechanisms for the limitation of the accumulation of autocatalytically produced HBrO₂ (one of the internal species) during the course of this reaction.     

A survey and new results on computer enumeration of polyhex and fusene hydrocarbons

After a short historic review, we briefly describe a new algorithm for constructive enumeration of polyhex and fusene hydrocarbons. In this process our algorithm also enumerates isomers and symmetry groups of molecules (which implies enumeration of enantiomers). Contrary to previous methods often based on the boundary code or its variants (which records orientation of edges along the boundary) or on the DAST code, which uses a rigid dualist graph (whose vertices are associated with faces and edges with adjacency between them), the proposed algorithm proceeds in two phases. First inner dual graphs are enumerated; then molecules obtained from each of them by specifying angles between adjacent edges are obtained. Favorable computational results are reported. The new algorithm is so fast that output of the structures is by far the most time-consuming part of the process. It thus contributes to enumeration in chemistry, a topic studied for over a century, and is useful in library making, QSAR/QSPR, and synthesis studies.     

Details of the reaction graphs for intramolecular isomerizations of the carboranes

From proposed mechanisms for framework reorganizations of the carboranes C₂B _n-2H _n ,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.     

Molecular structures of organoelement compounds and their representation as labeled molecular hypergraphs

Representation of molecular structures of organoelement compounds as hypergraphs with different degrees of detailing and different types of labels is considered. As opposed to the graph model, the hypergraph model contains not only ordinary edges connecting pairs of vertices and corresponding to ordinary covalent bonds but also hyperedges, each connecting more than two vertices and corresponding to delocalized multicenter bonds. Symbolic, numerical, and structural labels are proposed as labels of hypergraph vertices and edges. Structural labels involve substitutions and transformations on the sets of vertices and edges of a hypergraph. Choosing a method of representation is determined by conditions of a particular problem. Institute of Mathematics, Siberian Branch, Russian Academy of Sciences (Novosibirsk). Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 2, pp. 328–337, March–April, 1998.     

Computer generation of acyclic graphs based on local vertex invariants and topological indices. Derived canonical labelling and coding of trees and alkanes

A new procedure (GENLOIS) is presented for generating trees or certain classes of trees such as 4-trees (graphs representing alkanes), identity trees, homeomorphical irreducible trees, rooted trees, trees labelled on a certain vertex (primary, secondary, tertiary, etc.). The present method differs from previous procedures by differentiating among the vertices of a given parent graph by means of local vertex invariants (LOVIs). New graphs are efficiently generated by adding points and/or edges only to non-equivalent vertices of the parent graph. Redundant generation of graphs is minimized and checked by means of highly discriminating, recently devised topological indices based either on LOVIs or on the information content of LOVIs. All trees onN + 1 (N + 1 < 17) points could thus be generated from the complete set of trees onN points. A unique cooperative labelling for trees results as a consequence of the generation scheme. This labelling can be translated into a code for which canonical rules were recently stated by A.T. Balaban. This coding appears to be one of the best procedures for encoding, retrieving or ordering the molecular structure of trees (or alkanes).Dedicated to Professor Alexandru T. Balaban on the occasion of his 60th anniversary.     

Graphs whose vertices are graphs with bounded degree: Distance problems

By an f-graph we mean a graph having no vertex of degree greater than f. Let U(n,f) denote the graph whose vertex set is the set of unlabeled f-graphs of order n and such that the vertex corresponding to the graph G is adjacent to the vertex corresponding to the graph H if and only if H is obtainable from G by either the insertion or the deletion of a single edge. The distance between two graphs G and H of order n is defined as the least number of insertions and deletions of edges in G needed to obtain H. This is also the distance between two vertices in U(n,f). For simplicity, we also refer to the vertices in U(n,f) as the graphs in U(n,f). The graphs in U(n,f) are naturally grouped and ordered in levels by their number of edges. The distance nf/2 from the empty graph to an f-graph having a maximum number of edges is called the height of U(n,f). For f =2 and for f≥(n-1)/2, the diameter of U(n,f) is equal to the height. However, there are values of the parameters where the diameter exceeds the height. We present what is known about the following two problems: (1) What is the diameter of U(n,f) when 3≥f<(n-1)/2? (2) For fixed f, what is the least value of n such that the diameter of U(n,f) exceeds the height of U(n,f)?     

Topicity of vertices and edges in the möbius ladders: a topological result with chemical implications

It is shown that for uncolored Möbius ladders with n rungs, if n=3 all vertices and edges are constitutionally equivalent, while if n>3 all vertices are equivalent but two heterotopic classes of edges appear. Possible chemical consequences of these facts are discussed.     

Resonance in catacondensed benzenoid hydrocarbons

We consider resonance in cata-condensed benzenoids having six and seven fused benzene rings. The resonance relationship between the Kekule valence structures of the molecules is represented by the resonance graphs in which the vertices represent the Kekule valence structures, and the edges, the presence of the quantum chemical resonance integral involving permutation of three CC double bonds (within a single six-membered ring). The construction of the resonance graphs for large benzenoids is outlined and the properties of the derived resonance graphs are discussed. It is shown that the leading eigenvalue of the resonance graphs correlates with the resonance energy of benzenoids. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 585–600, 1997     

Stability of solutions to a reaction diffusion system based upon chemical reaction kinetics

In this paper, we deal with the stability problem to some mathematical models that describe chemical reaction kinetics. One is a set of ordinary differential equations induced by one reversible chemical reaction mechanism containing three chemical species. The other is a set of reaction diffusion equations based on the same chemical reaction. We show that all solutions of the model are asymptotically stable by applying the Liapunov method. We thus find that the concentration of each species has certain limits as time proceeds.      

Graph theoretical classification and coding of chemical reactions with a linear mechanism

Linear mechanisms of catalytic and noncatalytic chemical reactions which are theoretically feasible have been classified and coded using a detailed procedure for the unique numbering of cycles, edges, and vertices in the kinetic graphs. The following classification criteria are used in a hierarchical order: number of cycles and vertices, mutual connectivity of the cycles, manner of linking any pair of cycles, number of elements linking two cycles, mutual position of two cycles joined to a third one, orientation of edges, and presence of pendant vertices. All the types and classes of mechanisms are presented for reactions having up to five and four routes, respectively.     

On the equitability of multiply-connected monolayered cyclofusene

Multiply-connected monolayered cyclofusene (MMC) is a fused hexacyclic system with an exterior region and at least two interior empty regions called holes, as in figure 1. Each hexacyle has either: (a) two edges belonging to an exterior boundary and at least one hole, or (b) two edges belonging to boundaries of at least two holes. Let G be the graph of a given MMC. We show that G is equitable if and only if the set of vertices belonging to three hexacycles is equitable.     

Subcomponent Self‐Assembly of a 4 nm M4L6 Tetrahedron with ZnII Vertices and Perylene Bisimide Dye Edges

Formation of a tetrahedron with >4 nm perylene bisimide (PBI) dye edges and Zn^II vertices in a one‐pot 22 component self‐assembly reaction is reported. The luminescent polyhedron equilibrates to a Zn₂L₃ helicate and disassembles upon dilution. Insights into the subcomponent self‐assembly of extended PBI ligands help to refine design rules for constructing large photofunctional metallosupramolecular hosts.     

The enumeration of electron-rich and electron-poor polyhedral clusters

We present as dual processes the capping of closed triangulated polyhedra with apical atoms and the making of holes in such polyhedra either by puncture of the surface or by excision of atoms and their edges. These processes are shown to generate stable chemical species containing respectively less or more than 2n + 2 skeletal electrons. The former species are designated as electron-poor whereas the latter are called electron-rich. Pólya's enumeration method is used to enumerate the distinct ways of capping and excising the closed, triangulated polyhedra to yield systems containing from four to twelve vertices. For the enumeration of cappings the appropriate cycle index is that of the dual of the polyhedron being capped, whilst for the enumeration of the excisions the cycle index is that of the polyhedron being excised.     

Hamilton cycles and paths in fullerenes

It has been conjectured that every fullerene, that is, every skeleton of a spherical trivalent graph whose set of faces consists of pentagons and hexagons alone, is Hamiltonian. In this article the validity of this conjecture is explored for the class of leapfrog-fullerenes. It is shown that, given an arbitrary fullerene F, the corresponding leapfrog-fullerene Le(F) contains a Hamilton cycle if the number of vertices of F is congruent to 2 modulo 4 and contains a long cycle missing out only two adjacent vertices, and thus also a Hamilton path, if the number of vertices of F is divisible by 4.     

Computer program for finding all possible cycles in graphs

A new approach is presented for identifying all possible cycles in graphs. Input data are the total numbers of vertices and edges, as well as the vertex adjacencies using arbitrary vertex numbering. A homeomorphically reduced graph (HRG) is constructed by ignoring vertices of degree less than three. The algorithm is based on successive generation of possible edge-combinations in the HRG. If a combination yields a cycle, it is either printed or stored and then finally printed in a list of all possible cycles arranged in the order of increasing ring size. A unique numbering of the cycle is used. The computer program is listed and exemplified. Computing times are given.     

Topological aspects of chemically significant polyhedra

This paper considers both static and dynamic properties of chemically significant polyhedra. Static properties of polyhedra consider relationships between the numbers and degrees/sizes of polyhedral vertices, edges, and faces; polyhedral symmetries; and numbers of topologically distinct polyhedra of various types. Dynamic properties of polyhedra involve studies of polyhedral isomerizations from both macroscopic and microscopic points of view. Macroscopic aspects of polyhedral isomerization can be described by graphs called topological representations in which the vertices correspond to different permutational isomers and the edges to single degenerate polyhedral isomerization steps. Such topological representations are presented for isomerizations of polyhedra having five, six, and eight vertices. Microscopic aspects of polyhedral isomerizations arise from consideration of the details of polyhedral topology, such as the topological aspects of diamond-square-diamond processes. In this connection, Gale diagrams are useful for describing isomerizations of five- and six-vertex polyhedra, including the Berry pseudorotation of a trigonal bipyramid through a square pyramid intermediate and the Bailar or Ray and Dutt twists of an octahedron through a trigonal prism intermediate.