On the anti-Kekulé number of leapfrog fullerenes

The Anti-Kekulé number of a connected graph G is the smallest number of edges that have to be removed from G in such way that G remains connected but it has no Kekulé structures. In this paper it is proved that the Anti-Kekulé number of all fullerenes is either 3 or 4 and that for each leapfrog fullerene the Anti-Kekulé number can be established by observing finite number of cases not depending on the size of the fullerene.     

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.     

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.     

Kekuléan benzenoids

A Kekulé structure for a benzenoid or a fullerene $\Gamma $ is a set of edges $K$ such that each vertex of $\Gamma $ is incident with exactly one edge in $K$ , i.e. a perfect matching. All fullerenes admit a Kekulé structure; however, this is not true for benzenoids. In this paper, we develop methods for deciding whether or not a given benzenoid admits a Kekulé structure by constructing Kekulé structures that have a high density of benzene rings. The benzene rings of the Kekulé structure $K$ are the faces in $\Gamma $ that have exactly three edges in $K$ . The Fries number of $\Gamma $ is the maximum number of benzene rings over all possible Kekulé structures for $\Gamma $ and the set of benzene rings giving the Fries number is called a Fries set. The Clar number is the maximum number of independent benzene rings over all possible Kekulé structures for $\Gamma $ and the set of benzene rings giving the Clar number is called a Clar set. Our method of constructing Kekulé structures for benzenoids generally gives good estimates for the Clar and Fries numbers, often the exact values.     

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.     

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.     

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.     

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.     

The number of Kekulé structures of polyominos on the torus

Let G be a (molecule) graph. A perfect matching, or Kekulé structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekulé structures of a (molecule) graph is interest in chemistry, physics and mathematics. In this paper, we focus on some polyominos on the torus and obtain the explicit expressions on the number of the Kekulé structures of them.     

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.     

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.     

Global Forcing Number of Benzenoid Graphs

A global forcing set in a simple connected graph G with a perfect matching is any subset S of E(G) such that the restriction of the characteristic function of perfect matchings of G on S is an injection. The number of edges in a global forcing set of the smallest cardinality is called the global forcing number of G. In this paper we prove several results concerning global forcing sets and numbers of benzenoid graphs. In particular, we prove that all catacondensed benzenoids and catafused coronoids with n hexagons have the global forcing number equal to n, and that for pericondensed benzenoids the global forcing number is always strictly smaller than the number of hexagons.     

π-Electron currents in fixed π-sextet aromatic benzenoids

In view of different patterns of π-electron density currents in benzenoid aromatic compounds it is of interest to investigate the pattern of ring currents in various classes of compounds. Recently such a study using a graph theoretical approach to calculating CC bond currents was reported for fully benzenoid hydrocarbons, that is, benzenoid hydrocarbons which have either π-sextets rings or “empty” rings in the terminology of Clar. In this contribution we consider π-electron currents in benzenoid hydrocarbons which have π-electron sextets and C=C bonds fully fixed. Our approach assumes that currents arise from contributions of individual conjugated circuits within the set of Kekulé valence structures of these molecules.     

Zhang–Zhang polynomials of cyclo-polyphenacenes

The Zhang–Zhang polynomial (i.e., Clar covering polynomial) of hexagonal systems is introduced by H. Zhang and F. Zhang, which can be used to calculate many important invariants such as the Clar number, the number of Kekulé structures and the first Herndon number, etc. In this paper, we give out an explicit recurrence expression for the Zhang–Zhang polynomials of the cyclo-polyphenacenes, and determine their Clar numbers, numbers of Kekulé structures and their first Herndon numbers.     

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.     

Counterexamples to a proposed algorithm for Fries structures of benzenoids

The Fries number of a benzenoid is the maximum number of benzenoid hexagons over all of its Kekulé structures (perfect matchings), and a Fries canonical structure is a perfect matching that realises this maximum. A recently published algorithm claims to determine Fries canonical structures of benzenoids via iterated Hadamard products based on the adjacency matrix (Ciesielski et?al. in Symmetry 2:1390–1400, 2010). This algorithm is re-examined here. Convergence is typically rapid and often yields a single candidate perfect matching, but the algorithm can give an exponential number of choices, of which only a small number are canonical. More worryingly, the algorithm is found to give incorrect results for the Fries number for some benzenoids with as few as seven hexagonal faces. We give a combinatorial reformulation of the algorithm in terms of linear combinations of perfect matchings (with weights at each stage proportional to the products of weights of the edges included in a matching). In all the cases we have examined, the algorithm converges to a maximum-weight matching (or combination of maximum-weight matchings), and where the algorithm fails, either no best Fries matching is of maximum weight, or a best Fries matching is of maximum weight but a sub-optimal matching of the same weight is chosen.     

Computing the Kekulé structure count for alternant hydrocarbons

A fast computer algorithm brings computation of the permanents of sparse matrices, specifically, molecular adjacency matrices. Examples and results are presented, along with a discussion of the relationship of the permanent to the Kekulé structure count. A simple method is presented for determining the Kekulé structure count of alternant hydrocarbons. For these hydrocarbons, the square of the Kekulé structure count is equal to the permanent of the adjacency matrix. In addition, for alternant structures the adjacency matrix for N atoms can be written in such a way that only an N/2 × N/2 matrix need be evaluated. The Kekulé structure count correlates with topological indices. The inclusion of the number of cycles improves the fit. When comparing with previous results, the variance decreases 74%. The calculated standard heat of formation correlates with the logarithm of the Kekulé structure count. This heat increments 349 kJ/mol each time the Kekulé structure count increases by one order of magnitude. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002