Characteristic polynomials of alternant edge weighted linear chains with subsequent application to some linear poly (<Emphasis Type="Italic">p</Emphasis>-phenylene) graphs

The algorithms for the calculation of the characteristic polynomial coefficients for the alternant edge weighted graphs of linear chains have been developed. These algorithms have been utilized to calculate the characteristic polynomials of the linear poly (p-phenylene) and the methylene substituted linear poly (p-phenylene) compounds. These compounds are found to be important materials for electro-optical and electronic applications. The recurrence relation to obtain the sum of the CP coefficients (S_CP^N_r ){\left({S_{\rm CP}^{N_r }}\right)} of any such poly (p-phenylene) in terms of the respective sums of its lower analogs has been derived. The number of Kekulé valence structures (K) for such molecular graphs have been presented. Excellent linear correlations of logarithm of sum of CP coefficients (log  S_CP^N_r )({\rm log} \, {S_{\rm CP}^{N_r }}) and logarithm of Kekulé valence structures count (log K) with the ambient conditions density and bulk modulus of linear poly (p-phenylene) have also been found.     

π‐electron currents in polycyclic conjugated hydrocarbons: Coronene and its isomers having five and seven member rings

We have outlined novel graph theoretical model for computing π‐electron currents in π‐electron polycyclic conjugated hydrocarbons. We start with Kekulé valence structures of a polycyclic conjugated hydrocarbon and their conjugated circuits. To each 4n+2 conjugated circuits we assign counter clockwise current i and to each 4n conjugated circuit we assign clockwise current i. By adding the contributions from all conjugated circuits in a single Kekulé valence structure one obtains π‐electron current pattern for the particular Kekulé valence structure. By adding the conjugated circuit currents in all Kekulé valence structure one obtains the pattern of π‐electron currents for considered molecule. We report here π‐electron current patters for coronene and 17 its isomers, which have been recently considered by Balaban et al., obtained by replacing one or more pairs of peripheral benzene rings with five and seven member rings. Our results are compared with their reported π‐electron current density patters computed by ab initio molecular orbital (MO) computations and satisfactory parallelism is found between two so disparate approaches. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Relating Total π-Electron Energy and Resonance Energy of Benzenoid Molecules with Kekulé- and Clar-Structure-Based Parameters

Summary. Within classes of isomeric benzenoid hydrocarbons various Kekulé- and Clar-structure-based parameters (Kekulé structure count, Clar cover count, Herndon number, Zhang–Zhang polynomial) are all mutually correlated. This explains why both the total π-electron energy (E), the Dewar resonance energy (DRE), and the topological resonance energy (TRE) are well correlated with all these parameters. Nevertheless, there exists an optimal value of the variable of the Zhang–Zhang polynomial for which it yields the best results. This optimal value is negative-valued for E, around zero for TRE, and positive-valued for DRE. A somewhat surprising result is that TRE and DRE considerably differ in their dependence on Kekulé- and Clar-structure-based parameters.     

Relating Total π-Electron Energy and Resonance Energy of Benzenoid Molecules with Kekulé- and Clar-Structure-Based Parameters

Within classes of isomeric benzenoid hydrocarbons various Kekulé- and Clar-structure-based parameters (Kekulé structure count, Clar cover count, Herndon number, Zhang–Zhang polynomial) are all mutually correlated. This explains why both the total π-electron energy (E), the Dewar resonance energy (DRE), and the topological resonance energy (TRE) are well correlated with all these parameters. Nevertheless, there exists an optimal value of the variable of the Zhang–Zhang polynomial for which it yields the best results. This optimal value is negative-valued for E, around zero for TRE, and positive-valued for DRE. A somewhat surprising result is that TRE and DRE considerably differ in their dependence on Kekulé- and Clar-structure-based parameters.     

Algebraic Structure Count of Some Cyclic Hexagonal-Square Chains on the Möbius Strip

The concept of ASC (Algebraic structure count) is introduced into theoretical organic chemistry by Wilcox as the difference between the number of so-called “even” and “odd” Kekulé structures of a conjugated molecule. Precisely, algebraic structure count (ASC-value) of the bipartite graph G corresponding to the skeleton of a conjugated hydrocarbon is defined by where A is the adjacency matrix of G. The determination of algebraic structure count of (bipartite) cyclic hexagonal-square chains in the the class of plane such graphs is known. In this paper we expand these considerations on the non-plane class. An explicit combinatorial formula for ASC is deduced in the special case when all hexagonal fragments are isomorphic.     

A combination of Clar number and Kekulé count as an indicator of relative stability of fullerene isomers of C<Subscript>60</Subscript>

Kekulé count is not as useful in predicting the thermodynamic stability of fullerenes as it is for benzenoid hydrocarbons. For example, the Kekulé count of the icosahedral C₆₀, the most stable fullerene molecule, is surpassed by its 20 fullerene isomers (Austin et al. in Chem Phys Lett 228:478–484, 1994). This article investigates the role of Clar number in predicting the stability of fullerenes from Clar’s ideas in benzenoids. We find that the experimentally characterized fullerenes attain the maximum Clar numbers among their fullerene isomers. Our computations show that among the 18 fullerene isomers of C₆₀ achieving the maximum Clar number (8), the icosahedral C₆₀ has the largest Kekulé count. Hence, for fullerene isomers of C₆₀, a combination of Clar number and Kekulé count predicts the most stable isomer.     

Pairing theorem of graph eigenvalues: Its new proof and a generalization

Each undirected graph has its own adjacency matrix, which is real and symmetric. The negative of the adjacency matrix, also real and symmetric, is a well-defined mathematically elementary concept. By this negative adjacency matrix, the negative of a graph can be defined. Then an orthogonal transformation can be readily found that transforms a negative of an alternant graph to that alternant graph: (?G) → G. Since the procedure does not involve the edge weights, the pairing theorem holds true for all edge-weighted alternant graphs, including the usual “standard” 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.     

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.     

Ordering of Kekulé structures using nonadjacent numbers

Kekulé indices and conjugated circuits are computed for 36 Kekulé structures, together with two VB quantities associated with the corresponding factor graphs (previously called submolecules). These latter quantitites are nonadjacent numbers of Hosoya and the reciprocal of the connectivity indices of Randi?. It was found that the index of Hosoya successfully orders a set of Kekulé structures belonging to the same hydrocarbon in a parallel order as their Kekulé indices and branching indices. This substantiates the relation between VB and MO theories. A code is derived by summing contributions of nonadjacent numbers in all Kekulé stuctures of a hydrocarbon. The order of the resulting codes is found to be identical to the order of the molecular properties (resonance energies, π-energies, and eigenvalues) of the hydrocarbons.     

Topological hamiltonian spectra of polycyclic benzenoid hydrocarbons

A general theory which points out the relations between Hückel-electron energy, the number of Kekulé structures and the HOMO-LUMO separation is presented. Some normalized topological invariants are derived from the concept of the spectral density function. A reasonably simple (three parameters) model spectral density function leads to universal relations between topological invariants that, although valid for any alternant molecule, were tested numerically for polycyclic benzenoid hydrocarbons. Some general conclusions concerning a distribution of the adjacency matrix eigenvalues are drawn.     

Algebraic Kekulé formulas for benzenoid hydrocarbons

By assigning two pi-electrons of CC double bonds in a Kekulé valence structure to a benzene ring if not shared by adjacent rings and one pi-electron if CC double bond is shared by two rings we arrived at numerical valence formulas for benzenoid hydrocarbons. We refer to numerical Kekulé formulas as algebraic Kekulé valence formulas to contrast them to the traditional geometrical Kekulé valences formulas. The average over all numerical Kekulé valence structures results in a single numerical structure when a benzenoid hydrocarbon molecule is considered. By ignoring numerical values the novel quantitative formula transforms into a qualitative one which can replace incorrectly used notation of pi-electron sextets to indicate aromatic benzenoids by placing inscribed circles in adjacent rings-which contradicts Clar's characterization of benzenoid hydrocarbons.     

Innate degree of freedom of a graph

For a Kekulé structure we consider the smallest number of placements of double bonds such that the full Kekulé structure on the given parent graph is fully determined. These numbers for each Kekulé structure of the parent graph sum to a novel structural invariant F, called the degree of freedom of the graph. Some qualitative characteristics are identified, and it is noted that apparently it behaves differently from a couple of other invariants related to Kekulé structures.     

A fast computer algorithm for finding the permanent of adjacency matrices

A fast computer algorithm is described which brings computation of the permanents of sparse matrices, specifically, chemical adjacency matrices, within the reach of a desktop computer. Examples and results are presented, along with a discussion of the relationship of the permanent to the Kekulé structure count. Also presented is a C-language implementation which was deliberately written for ease of translation into other high-level languages.     

Stability of Conjugated Carbon Nanocones

We use interlacing techniques to prove that carbon nanocones who have a Fries Kekulé structure have closed Hückel shells, and that this result can be extended to all conjugated cones where each edge belongs to a hexagonal face and the configuration of the non-hexagonal faces are consistent with a Fries Kekulé structure. Cones with Fries Kekulé structure or substructure are topical—not only from a valence bond theoretical point of view—since a previous ab initioanalysis favored cones where the pentagons at the tip are configured as in a Fries Kekulé structure. The question of interdependence will therefore be addressed.     

Kekulé’s Oscillating D3h Cyclohexatriene Structure of Benzene

Kekulé postulated that neighbouring carbon atoms in benzene undergo incessant collision with each other, thereby leading to the interchange of double and single bonds, which amounts to an oscillation between two cyclohexatriene structures in dynamic equilibrium. It has been claimed that Kekulé arrived at a fully symmetric D_6h structure of benzene and that the oscillation hypothesis should not be attributed to him. However, Clausius’ collision theory, which was known at the time, implies that, when the absolute temperature approaches zero, the collision frequency will tend toward zero too, i.e. collisions will stop, and a static, D_3h cyclohexatriene obtains. The classical collision theory did not allow Kekulé to construct the desired D_6h structure as the energy minimum. The theory of harmonic oscillators would have allowed it, but that was not attempted at Kekulé’s time.     

The number of Kekulé structures and the thermodynamic stability of conjugated systems

The dependence of Hückel π-electron energies, E_π, on the basic graph theoretical parameters N (the number of vertices), ν (the number of edges) and ASC (the algebraic structure count) is explored. The form with the ASC enters E_π is established and an equation for E_π is developed. It is shown how the early and apparent success of the (resonance) theory rested on the fortunate fact that all Kekulé structures for benzenoid hydrocarbons and acyclic polyenes have the same parity. The significance of ASC in determining chemical stability and reactivity is dicussed briefly.     

A combinatorial/geometric analysis of convex cyclofusene

Previously, we presented an algorithm for counting Kekulé structures for parallelogram-like benzenoids with holes by counting descending paths using rectangular meshes with holes. In this article, we describe an algorithm to count Kekulé structures for convex cyclofusenes using a combinatorial/geometric approach.     

Bond length features of linear carbon chains of finite to infinite size: Visual interpretation from Pauling bond orders

Schemes for Kekulé structure counting of linear carbon chains are suggested. Mathematical formulas, which calculate the Pauling bond order P(k, N) of a chemical bond numbered by k, are given for the carbon chain with N carbon atoms. By use of the least‐squares fitting of a linearity, relationships between Pauling bond orders and bond lengths are obtained, and such correlation of the Pauling bond order–bond length can be qualitatively extended to the excited states. The relative magnitudes of Pauling bond orders in unsaturated carbon chains dominate C–C bond lengths a well as the bond length feature with the chain size increasing. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 144–149, 2003