Kekule Structures Research Articles

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

An Upper Bound for the Clar Number of Fullerene Graphs

A fullerene graph is a three-regular and three-connected plane graph exactly 12 faces of which are pentagons and the remaining faces are hexagons. Let F n be a fullerene graph with n vertices. The Clar number c(F n ) of F n is the maximum size of sextet patterns, the sets of disjoint hexagons which are all M-alternating for a perfect matching (or Kekule structure) M of F n . A sharp upper bound of Clar number for any fullerene graphs is obtained in this article: $$c(F_n)\leqslant \lfloor \frac {n-12} 6\rfloor$$ . Two famous members of fullerenes C60 (Buckministerfullerene) and C70 achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.

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” Kekule 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 \({\rm ASC} \{G \} \mathop{=}\limits^{\rm def} \sqrt{\mid {\rm det} A \mid}\) 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.

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Huckel molecular graphs with Kekule structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by \(\mathcal{H}_n^l\) such that for any graph in \(\mathcal{H}_n^l\), n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of \(\mathcal{H}_n^l\) have been discussed.

K-resonance in Toroidal Polyhexes*

This paper considers the k -resonance of a toroidal polyhex (or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). A toroidal polyhex G is said to be k-resonant if, for 1≤ i ≤ k, any i disjoint hexagons are mutually resonant, that is, G has a Kekule structure (perfect matching) M such that these hexagons are M-alternating (in and off M). Characterizations for 1, 2 and 3-resonant toroidal polyhexes are given respectively in this paper.

Stability of Conjugated Carbon Nanocones

We use interlacing techniques to prove that carbon nanocones who have a Fries Kekule structure have closed Huckel 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 Kekule structure. Cones with Fries Kekule 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 Kekule structure. The question of interdependence will therefore be addressed.

A comparison between 1-factor count and resonant pattern count in plane non-bipartite graphs

The concept of resonant (or Clar) pattern is extended to a plane non-bipartite graph G in this paper: a set of disjoint interior faces of G is called a resonant pattern if such face boundaries are all M-conjugated cycles for some 1-factor (Kekule structure or perfect matching) M of G. In particular, a resonant pattern of benzenoids and fullerenes coincides with a sextet pattern. By applying a novel approach, the principle of inclusion and exclusion in combinatorics, we show that for any plane graphs, 1-factor count is not less than the resonant pattern count, which generalize the corresponding results in benzenoid systems and plane bipartite graphs. Applications to fullerenes are also discussed.

On the number of Kekulé structures in capped zigzag nanotubes

Two theoretical formulae for the number of Kekule structures in general capped zigzag nanotubes are established: one of which is by using the techniques of the transfer matrices, the other involves the eigenvalues of the transfer matrix which reveals the asymptotic behaviour of this index. In effective, according to the symmetric aspect of the tubule, the order of the transfer matrix could be notably decreased. As an application, the closed expressions for four types are given out and the relevant numerical results for those of length up to 50 are listed.

An Alternative Strategy for Count and Storage of Kekule and Longer Range Resonance Valence Bond Structures.

In this paper, we suggest a simple representation for notation of Kekule valence bond (KVB) structures and longer range resonance valence bond (RVB) structures, which is called adjacency bonding array. In this representation, only an N component one-dimensional array is needed for inscribing each KVB or longer range RVB structure for an N-carbon system. Based on the adjacency bonding arrays, we develop very efficient algorithms for the systematic search of KVB and RVB structures as well as evaluation of the basis set overlap and Hamiltonian matrices.

Partitioning of π -electrons in rings of polycyclic conjugated hydrocarbons: Part 6. Comparison with other methods for estimating the local aromaticity of rings in polycyclic benzenoids

By adopting the convention that shared double bonds in polycyclic conjugated hydrocarbons contribute with one π-electron and unshared ones with two π-electrons, a partition of π-electrons in each ring (π-electron content, EC) can be obtained by averaging over all Kekule structures, which are assumed to have equal weights. This affords a simple measure of local aromaticity that is comparable with other such local aromaticity indices in polycyclic benzenoids.

An ab Initio Valence Bond Study on Cyclopenta-Fused Naphthalenes and Fluoranthenes

To probe the effect of external cyclopenta-fusion on a naphthalene core, ab initio valence bond (VB) calculations have been performed, using strictly atomic benzene p-orbitals and p-orbitals that are allowed to delocalize, on naphthalene (1), acenaphthylene (2), pyracylene (3), cyclopenta[b,c]acenaphthylene (4), fluoranthene (5), and cyclopenta[c,d]fluoranthene (6). For the related compounds 1-4 and 5,6 the total resonance energies (according to Pauling's definition) are similar. Partitioning of the total resonance energy in contributions from the possible 4n + 2 and 4n pi-electron conjugated circuits shows that only the 6pi-electron conjugated circuits (benzene-like) contribute to the resonance energy. The results show that cyclopenta-fusion does not extend the pi system in the ground state; the five-membered rings act as peri-substituents. As a consequence, the differences in (total) resonance energy do not coincide with the differences in thermodynamic stability. Notwithstanding, the relative energies of the Kekule structures can be estimated using Randic's conjugated circuits model.

On the Distribution of π-Electrons into Rings of Conjugated Hydrocarbons Containing a Linear Polyacene Fragment

A method for assessing the π-electron contents (EC) of rings of benzenoid hydrocarbons, based on the examination of their Kekule structures, was recently put forward by Balaban and Randic. We now show that all hexagons belonging to a linear polyacene fragment of a conjugated hydrocarbon (not necessarily benzenoid) have mutually equal EC-values.

Molecules‐in‐Molecule Estimation of the Extent of Localization of Kekulean Substructures in Polycyclic Aromatic Hydrocarbons.

This article first revises graph-theoretical (local aromaticity and overall molecular) indices, introduced by M. Randic in 1975, for benzenoid hydrocarbons and somewhat improves them for computer enumeration. This goes beyond total Kekule structure enumeration, yielding an index calculation useful for the quantitative estimation of localization of different Kekulean substructures (including ethylene-, benzene-, annulene-, and radialene-units). This may be viewed as a “molecules-in molecule” approach to polycyclic aromatic hydrocarbons within the context of graph theoretical partitioning.

Characterization of distribution of pi-electrons amongst benzenoid rings for Randić’s “algebraic” Kekulé structures

For benzenoid hydrocarbons the distribution of pi-electrons amongst rings is characterized in the context of Randic’s mode of assignment attending to the different Kekule structures. In particular the mean and mean deviation from the mean are considered, and the benzenoids which achieve maximum deviation are identified.

Cyclic Conjugation in Benzo-Annelated Perylenes. How Empty is the “Empty” Ring?

Cyclic conjugation in benzo-annelated perylenes is studied by means of the energy-effects of their six-membered rings. “Classical” theoretical approaches (based on Kekule structures, Clar formulae, or conjugated circuits) predict that the central ring in benzo-annelated perylenes is “empty” and thus negligibly contributes to cyclic conjugation. Our calculations show that this is true only to a limited degree. In particular, rings angularly annelated relative to the central ring significantly increase the extent of its cyclic conjugation.

Partitioning of π-electrons in Rings of Polycyclic Conjugated Hydrocarbons. Part 4. Benzenoids with More Than One Geometric Kekulé Structure Corresponding to the Same Algebraic Kekulé Structure

A rigorous proof is provided for the theorem on the necessary and sufficient conditions for the algebraic Kekule structure of a peri-condensed or corona-condensed benzenoid hydrocarbon to correspond to more than one geometric Kekule structure.

Coding and Ordering Kekule Structures.

The concept of numerical Kekule structures is used for coding and ordering geometrical (standard) Kekule 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 Kekule structures can be obtained for any class of polycyclic conjugated systems that possesses standard Kekule structures. The reconstruction of standard Kekule 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 Kekule structures: the Wiswesser code for the benzenoid and the numerical code for its Kekule structure. There is an additional problem with pericondensed benzenoid hydrocarbons; there appear numerical Kekule structures that correspond to two (or more) geometrical Kekule structures. However, this problem can also be resolved.

Algebraic Kekule Structures of Benzenoid Hydrocarbons.

ChemInformVolume 35, Issue 24 Other Subjects Algebraic Kekule Structures of Benzenoid Hydrocarbons. Ivan Gutman, Ivan Gutman Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this authorDamir Vukicevic, Damir Vukicevic Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this authorAnte Graovac, Ante Graovac Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this authorMilan Randic, Milan Randic Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this author Ivan Gutman, Ivan Gutman Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this authorDamir Vukicevic, Damir Vukicevic Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this authorAnte Graovac, Ante Graovac Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this authorMilan Randic, Milan Randic Fac. Sci., Univ. Split, HR-21000 Split, CroatiaSearch for more papers by this author First published: 19 May 2004 https://doi.org/10.1002/chin.200424201AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume35, Issue24June 15, 2004 RelatedInformation

K-Resonance of Open-Ended Carbon Nanotubes

An open-ended carbon nanotube or a tubule is a part of some regular hexagonal tessellation of a cylinder. A tubule T is said to be k-resonant if for every k (or fewer) pairwise disjoint hexagons, the subgraph obtained from T by deleting all the vertices of these hexagons must have a Kekule structure (perfect matching) or must be empty. The 1-resonant tubules can be constructed by an approach provided in H. Zhang and F. Zhang, Discrete Appl. Math. 36 (1992) 291. In this paper, we give the construction method of k(k≥3)-resonant tubules. The lower bound of its Clar number of k(k≥3)-resonant tubules is also given. Note that the present paper does not consider the capped species.

The Gutman Formulas for Algebraic Structure Count

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” Kekule 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 \(ASC\{ G\} = \sqrt {|\det A|} \) where A is the adjacency matrix of G. In the case of bipartite planar graphs containing only circuits of the length of the form 4s+2 (s=1,2,...) (the case of benzenoid hydrocarbons), this number is equal to the number of the perfect matchings (K-value) of G. However, if some of circuits are of the length 4s (s=1,2,...) then the problem of evaluation ASC-value becomes more complicated. The theorem formulated and proved in this paper gives a simple and efficient algorithm for calculation of algebraic structure count of an arbitrary bipartite graphs with n+n vertices. Three recurrence formulas for the algebraic structure count – the Gutman formulas, which are closely analogous to the well-known recurrence formula K{G}=K{G−e}+K{G−(e)} for the number of perfect matchings (G−e is the subgraph obtained from the graph G by deleting the edge e and G−(e) is the subgraph obtained from G by deleting both the edge e and its terminal vertices) are obtained as a simple corollary of the theorem.