Recursive Formulae for Enumeration of LM-Conjugated Circuits in Structurally Related Benzenoid Hydrocarbons

The linearly independent and minimal conjugated (LM-conjugated) circuits of benzenoid hydrocarbons play the central role in the conjugated circuit model. For a general case, the enumeration of LM-conjugated circuits may be tedious as it requires construction of all Kekule structures. In our previous work, a recursive method for enumeration of LM-conjugated circuits of benzenoid hydrocarbons was established. In this paper, we further extend the recursive formulae for enumerations of LM-conjugated circuits for both catacondensed benzenoid hydrocarbons and some families of structurally related pericondensed benzenoid hydrocarbons.     

Benzenoid rings resonance energies and local aromaticity of benzenoid hydrocarbons

In this article, we consider partitioning of the analytical expression for resonance energy (RE) in smaller benzenoid hydrocarbons, to individual benzenoid rings of polycyclic molecules. The analytical expression for molecular RE, available since 1976, is given by the count of all linearly independent conjugated circuit in all Kekulé structures in a molecule. Analytical expression for local ring RE (RRE) is given by counting all linearly independent conjugated circuits involving single benzenoid ring in all Kekulé structures, which when added, gives the molecular RE. If for benzene ring the RRE is taken to be 1.000, rings in polycyclic benzenoid hydrocarbons have their ring RRE, which give the degree of their local aromaticity, smaller than 1.000. The difference to 1.000 is a measure of the similarity of a ring to benzene in this one-dimensional (1-D) representation of local aromaticities of benzenoid hydrocarbons. The plot of RRE against the distance of the same ring from benzene in the Local Aromaticity Map, in which benzenoid rings are characterized ring bond orders and average variations of adjacent CC bonds, shows linear correlation (with r = 0.91), reducing the local aromaticity in benzenoid hydrocarbons to 1-D molecular property. © 2018 Wiley Periodicals, Inc.     

Perimeter topology of benzenoid polycyclic hydrocarbons

The equivalence of the perimeter topological equations derived by Dias and the 13 possible modes of hexagon adjacency in fused benzenoid systems subsequently presented by Cyvin, Gutman, and collaborators is shown. The aufbau principle for generation of all fused total resonant sextet (TRS) benzenoid hydrocarbons is proved. The precise distinction and topological properties between fused, nonfused, strain-free, strained, helical, and nonhelical benzenoid hydrocarbons are carefully delineated. The TRS benzenoid isomers have the maximum number of bay regions, and expressions for the number of bay regions that any given TRS benzenoid will have are given.     

An efficient algorithm for generating all Kekulé patterns of a generalized benzenoid system

An algorithm for generating all Kekulé patterns of a benzenoid system was given by Jiang [1]. However, for a generalized benzenoid system, this problem is more complex. In this paper, we give an efficient algorithm for generating all Kekulé patterns of a generalized benzenoid system by the generalized directed tree structure [2] of the set of Kekulé patterns of a generalized benzenoid system.Project supported by NSFC.     

On isospectral benzenoid graphs

The recently Proposed procedure [5] for the construction of isospectral benzenoid graphs has been examined in detail. Necessary and sufficient conditions for the construction of isospectral benzenoid graphs with isomorphicH-graphs are formulated. The inapplicability of the Procedure for the construction of isospectral benzenoid graphs with an even number of vertices has been proven.     

Total resonant sextet benzenoid hydrocarbon isomers and their molecular orbital and thermodynamic characteristics

The more stable strain-free total resonant sextet isomers of benzenoid hydrocarbons are predicted to be probable combustion pollutants. These benzenoid hydrocarbons have been enumerated, and their topological characteristics examined. Total resonant sextet benzenoid isomers make up less than 1% of all the benzenoid hydrocarbons having formulas with N_c divisible by 6. There is a thermodynamic preference for the larger and more highly condensed polycyclic aromatic hydrocarbon molecule. The Hückel molecular orbital (HMO) and thermodynamic parameters of these total resonant sextet benzenoid hydrocarbons have been computed and tabulated.     

Normal components of benzenoid systems

Summary Consider a benzenoid system with fixed bonds and the subgraph obtained by deleting fixed double bonds together with their end vertices and fixed single bonds without their end vertices. It has often been observed for particular benzenoid systems, and conjectured (or stated) that, in general, such a subgraph has at least two components, and that each component is also a benzenoid system and is normal. But there are no rigorous proofs for that. The aim of this paper is to present mathematical proofs of those two facts. It is also shown that if a benzenoid system has a single hexagon as one of its normal components then it has at least three normal components.     

Benzenoid links

In this paper, we study new configurations of benzenoid hydrocarbons, called benzenoid links. Roughly speaking, a primitive corofusene is a closed narrow hexagonal ribbon with out-of-plane curvature 0. A primitive corofusene or the union of disjoint primitive corofusenes in \mathbbR³{\mathbb{R}^{3}} is called a benzenoid link. In this paper, we determine the minimum number of hexagons needed for a nontrivial benzenoid link in different senses. We also determine the structures of the smallest and the second smallest nontrivial benzenoid links of different types and their numbers of Kekule structures. We list all the benzenoid Hopf links of type III with 22–25 hexagons by their canonical codes in the appendix.     

Determining the number of resonance structures in concealed non-Kekuléan benzenoid hydrocarbons

The number of resonance structures (SC) for previously published concealed non-Kekuléan benzenoid hydrocarbons is determined. Using a simple computer program, analytical expressions for determining SC for various classes of non-Kekuléan (free-radical) benzenoid hydrocarbons are derived, and some properties of concealed non-Kekuléan benzenoid hydrocarbons are studied.     

Polycyclische benzenoide Systeme

Superposition of significant electron structures is used to estimate the mesomerism stabilization of polycyclic benzenoid systems by consideration of the number ofKekulé structures and of benzenoid interactions between them. General formulae and stability rules are given for many classes of cata and pericondensed molecules. The decoupling of large benzenoid systems into isolated subsystems is described. Three distinct types of polymeric benzenoid systems exist: unstable systems with vanishing mesomerism energy per monomeric umit, and stable systems with nonvanishing mesomerism energy, which are or are not decoupled into isolated subsystems.

     

k-resonant benzenoid systems and k-cycle resonant graphs.

A benzenoid system (or hexagonal system) H is said to be k-resonant if, for 1 < or = t < or = k, any t disjoint hexagons of H are mutually resonant; that is, there is a Kekule structure (or perfect matching) K of H such that each of the k hexagons is an K-alternating hexagon. A connected graph G is said to be k-cycle resonant if, for 1 < or = t < or = k, any t disjoint cycles in G are mutually resonant. The concept of k-resonant benzenoid systems is closely related to Clar's aromatic sextet theory, and the concept of k-cycle resonant graphs is a natural generalization of k-resonant benzenoid systems. Some necessary and sufficient conditions for a benzenoid system (respectively a graph) to be k-resonant (respectively k-cycle resonant) have been established. In this paper, we will give a survey on investigations of k-resonant benzenoid systems and k-cycle resonant graphs.     

Analytical expressions for the count of LM-conjugated circuits of benzenoid hydrocarbons

A recursive method for enumeration of linearly independent and minimal conjugated circuits of benzenoid hydrocarbons had previously been given which is valid for several classes of benzenoid hydrocarbons. In the present article, the properties and constructions of unique minimal conjugated circuits and pairs of minimal conjugated circuits of a ring s in a benzenoid hydrocarbon B are investigated. An analytical expression for the count of LM-conjugated circuits of B is given which is based on the counts of Kekulé structures of selected subgraphs of B. By using the method, the LMC expression of any benzenoid hydrocarbon can be obtained. © 1996 John Wiley & Sons, Inc.     

On the Anti-forcing Number of Benzenoids

The anti-forcing number is introduced as the smallest number of edges that have to be removed that any benzenoid remains with a single Kekulé structure. Similarly, the anti- Kekulé number is discussed as the smallest number of edges that have to be removed that any benzenoid remains connected but without any Kekulé structure. These concepts have been exemplified on damaged benzenoid parallelograms.     

Graph theoretical characterization and computer generation of certain carcinogenic benzenoid hydrocarbons and identification of bay regions

An algorithm is developed for generating and characterizing carcinogenic catacondensed benzenoid hydrocarbons. The bay regions in these structures are identified by a technique that we developed at Johns Hopkins. Using the three-digit code proposed by Balaban, and the concept of ring adjacency matrix expounded here, we generate catacondensed benzenoid hydrocarbons in the computer and identify the number of potentially carcinogenic bay regions in each of them. The results of computer generation agree with the combinatorial enumeration of Harary and Read. All structures containing up to five rings and some with six rings and the number of bay regions in these are presented. Computer results for the structures and bay regions of all seven-, eight-, and nine-membered unbranched catacondensed benzenoid hydrocarbons and the number of bays are available from the authors.     

Hamiltonian circuits,Hamiltonian paths and branching graphs of benzenoid systems

A benzenoid systemH is a finite connected subgraph of the infinite hexagonal lattice with out cut bonds and non-hexagonal interior faces. The branching graphG ofH consists of all vertices ofH of degree 3 and bonds among them. In this paper, the following results are obtained:

1. A necessary condition for a benzenoid system to have a Hamiltonian circuit.
2. A necessary and sufficient condition for a benzenoid system to have a Hamiltonian path.
3. A characterization of connected subgraphs of the infinite hexagonal lattice which are branching graphs of benzenoid systems.
4. A proof that if a disconnected subgraph G of the infinite hexagonal lattice given along with the positions of its vertices is the branching graph of a benzenoid system H, then H is unique.

     

The zero-field splitting parameter Dof the triplet state T1 of benzenoid hydrocarbons described in terms of basic molecular subunits

The zero-field splitting parameter D has been studied in dependence of the structure and topology of benzenoid hydrocarbons. A recent explanation in terms of localized benzenoid indices could be extended to larger molecular subunits. These have to be weighted according to the number of contributing Kekulé structures. Four rules for the determination of D could be formulated. For all known D-values of benzenoid hydrocabons (26 molecules) the theoretical D-values calculated according to the rules are in excellent agreement with the experimental data. With the method described the D-values of a great number of mulecules can be determined easily and accurately from the D-values of only few basic types of molecules.     

Coding and ordering Kekulé structures

The concept of numerical Kekulé structures is used for coding and ordering geometrical (standard) Kekulé structures of several classes of polycyclic conjugated molecules: catacondensed, pericondensed, and fully arenoid benzenoid hydrocarbons, thioarenoids, and [N]phenylenes. It is pointed out that the numerical Kekulé structures can be obtained for any class of polycyclic conjugated systems that possesses standard Kekulé structures. The reconstruction of standard Kekulé structures from the numerical ones is straightforward for catacondensed systems, but this is not so for pericondensed benzenoid hydrocarbons. In this latter case, one needs to use two codes to recover the geometrical Kekulé structures: the Wiswesser code for the benzenoid and the numerical code for its Kekulé structure. There is an additional problem with pericondensed benzenoid hydrocarbons; there appear numerical Kekulé structures that correspond to two (or more) geometrical Kekulé structures. However, this problem can also be resolved.     

Resonance energy of very large benzenoid hydrocarbons

The resonance energy of conjugated benzenoid systems is expressed as contributions arising from independent conjugated circuits. The scheme has been applied to numerous very large conjugated systems. In many cases, it was possible to find regularities in the increments for the resonance energy within a family of benzenoid systems as the number of benzene rings is increased.