Kekule Structures Research Articles

Overview of benzene and exploration of benzene structure

Benzene is a hydrocarbon the simplest aromatics. But the search for benzenes structure has been tortuous. This article mainly introduces the basic properties of benzene. From the physical properties of benzene to the chemical properties of benzene to the uses of benzene. In this work, we also summarize the process of scientists exploration of benzene structure and introduce the background of benzene. Of all the structures of benzene, the Kekule structure is the most recognized structure. We have proved the correctness of the Kekule structure by the product of the ozonation decomposition reaction of o-xylene. However, the Kekule structure has some limitations, the search for benzene structure is still ongoing, and we are still trying to find ways to better describe the structure of benzene. So we also sum up some modern theories about the structure of benzene. In addition, benzene is a very common chemical raw material. The article also lists some chemical products made from benzene.

Photoreaction of indazole metastable-state photoacid

Indazole metastable-state photoacids have been recently applied to modulating local pH of tumor and brain in animal studies. The mechanism of this type of photoacid was previously assumed to be the same as that of the extensively studied merocyanine photoacids. However, this work showed that the photo reaction of indazole photoacid is significantly different from merocyanine photoacid in water. An indazole photoacid with high water solubility up to 70 mM was designed and synthesized. Photoirradiation generates a quinoidal cis-conformer in water instead of the spiro conformer generally expected for a metastable-state photoacid. Although the twisted conformation reduced conjugation between the electron-withdrawing moiety and the acidic moiety, the cis-conformer possesses a higher acidity than that of the trans-conformer due to the unusual quinoidal structure. In organic solvents, the cis-isomer possesses a Kekule structure, which has lower acidity than that of the trans-conformer. In the presence of a weak base, photoirradiation reversibly converted the trans-conformer to the spiroconformer. Experiments showed that deprotonated cis-conformer is the intermediate for both the spiro formation and the reverse cis-to-trans reaction. Addition of an acid stops both reactions and locks the photoproduct to the protonated cis-conformer.

Acyclic Coloring of Certain Graphs

A graph G with acyclic coloring has no two adjacent vertices with the same color and no bichromatic cycle. Also, the coloring results in a forest when any two-color classes are combined. The concept of acyclic coloring plays a pivotal role in the computation of Hessians, Kekule structures classification, coding theory, and statistical mechanics. In this paper, the acyclic chromatic number of generalized fan graph, generalized Möbius ladder graph and flower snark graph have been determined.

Eccentricity‐based topological invariants of tightest nonadjacently configured stable pentagonal structure of carbon nanocones

AbstractConjugated open‐ended cones in which the configured pentagons are consistent, lies in the circle of Fries Kekule structure. This nonadjacent tightest configuration of pentagons as shown in Figure 1 is consistent with a Fries Kekule structure and thus provides the most stable cone. In the study, various topological indices of the same structure in regard of physico‐concoction resources and bioactivity of substance mixtures are studied which further helps to study the behavior of chemical compounds. In this regard, Zagreb indices and of a molecular graph G is used to evaluate the complexity in chemical systems and biological organisms. In this article, we consider two complex families of stable carbon nanocones and compute their ECI, TEI, and eccentricity‐based Zagreb indices.

The anti-Kekulé number of graphs

Perfect matchings in chemical graphs correspond to Kekule structures in underlying chemical structure, which play an important role in the analysis of resonance energy and stability of certain chemical compounds. In 2007, Vukicevic and Trinajstic introduced the concept of the anti-Kekule number to be the smallest number of edges that have to be removed from a graph such that it remains connected but without any perfect matching. Thereafter, this parameter attracted much more attention of researchers both in mathematics and theoretical chemistry. In the present paper, we report on the results related to the anti-Kekule of graphs. Some new results on the anti-Kekule number of nanotubes, polyomino system and cactus chains are also proven. We conclude the paper with some open problems.

Excessive Index of Certain Carbon Based Nanotubes

Chemical graph theory models have been extensively used as predictors of the properties of chemical compounds. Nanotechnology is the study of manipulating matter on an atomic and molecular scale. There are many applications of nanotechnology in the area of medicine, chemistry, energy, agriculture, information and communication, heavy industry, and consumer goods. A Kekule structure in a molecular graph is nothing but a perfect matching in the graph. The minimum number of Kekule structures that cover the edge set of a molecular graph G is known as the excessive index of G. In this article, we determine the excessive index for nanosheet, nanotube, nanotube, H-anthracenic nanotube, and H-tetracenic nanotube.

Kekule's Structure of Benzene and the Resonance Structures of Benzene

Kekule's Structure of Benzene and the Resonance Structures of Benzene

Opening a band gap in graphene by C–C bond alternation: a tight binding approach

In the present work, we consider the electronic properties of graphene with Kekule structure formed from two different C–C bonds in its hexagonal lattice. When the C–C bond alternation was introduced, a small band gap has been opened in the band structure of graphene and it increases linearly by a difference in the bond lengths δ. While the applied strain along the zigzag or armchair direction causes band gap to decrease rapidly to zero, the strain in the other directions can increase the band gap. Interestingly, when the graphene with Kekule structure is strained, its band gap is inversely proportional to the bond length difference δ. Opening a band gap in graphene due to bond alternation and strain can open up new applications in nanoelectronic devices.

Enumeration of Kekule Structures of Polyhex Monoradical Having Only Linear Assemblies of Benzenoid Hydrocarbons

A mathematical process of enumeration of Kekule structures of any polybenzenoid monoradical having only linear assemblies of benzenoid hydrocarbons is given by using a method which consisted of the excision of the radical center in order to obtain a conjugated hydrocarbon having a 12-annulene core ring appended to one or more polybenzenoid fragments, the partition of the resulting conjugated hydrocarbon into smaller independent resonant circuits. Polyhex monoradical skeletons are obtained from a building up procedure which consists to fuse in different ways the acenaphtyl moiety with one or more polybenzenoid fragments. Mathematical formulas are established for three cases of assemblies, one edge linear assemblies, two edges linear assemblies with two linear polybenzenoid fragments located at right angles θ = 60°, 120° or 180° and three edges linear assemblies with three linear polybenzenoid fragments located at right angles θ= 60°, 120° or 180°. Results show that the number of Kekule structures of any polybenzenoid monoradical having only linear assemblies of benzenoid hydrocarbons increases with the number of fused benzenoids according to results compiled in the table for h equal 1, 2, 3 and 4. Difficulty to represent graphs when the number of structures increases with the number of benzenoids cannot be neglected.

An extension of the Fries rule to non-benzenoid hydrocarbons having four-atomic rings

The study contains a deductive search for the principal monocyclic substructures determining different importances (weights) of separate Kekule valence structures of phenylenes and their congeners. Individual Kekule structures are modelled as continuous conjugated systems consisting of uniform double (C=C) bonds connected by uniform single (C–C) bonds, the latter being substantially weaker as compared to the former. The relevant total $$\pi$$ -electron energies are shown to offer an adequate criterion for ordering of the structures concerned according to their weights. These energies, in turn, are derived in the form of power series with respect to the small resonance parameter of C–C bonds. Analysis of expressions for separate members of this series shows that the cyclobutadienoid rings (if any) are the most important destabilizing contributors to the energy of the given structure, whereas the benzenoid rings and cycles like 3,4-dimethylene cyclobutene take the second place (their increments are stabilizing and destabilizing, respectively). Additivity and transferability of the above-enumerated contributions to energies of Kekule valence structures also are among the conclusions. These results provide us with an extension of the classical Fries rule to non-benzenoid hydrocarbons having four-atomic rings. Specific examples of the given class of compounds are considered in a detail, viz. biphenylene, [3]phenylene, as well as benzo- and naphthocyclobutenes. The relation of the approach applied to the theory of conjugated circuits also is discussed.

Magnetism-tuning strategies for graphene oxide based on magnetic oligoacene oxide patches model.

Graphene oxide (GO) has wide application potential owing to its 2D structure and diverse modification sites for various targeted uses. The introduction of magnetism into GO structures has further advanced the controllability of the application of GO materials. Herein, the concept of modular design and modeling was applied to tune the magnetism of GO. To obtain desirable magnetic properties, diradical-structured GO patches were formed by the introduction of two functional groups to break the Kekule structure of the benzene ring. In these diradical GO patches, the energy of the triplet state was lower than those of the open-shell broken-symmetry singlet state and closed-shell singlet state. To create such multi-radical patches, a practical approach is to determine a substantial spatial separation of the α and β spin densities in the molecule. Thus, systematic design strategies and tests were evaluated. The first strategy was extending the distance between the distribution center of the α and β spin densities; the second was controlling the delocalization directions of the α and β electrons; the third was controlling the delocalization extension of the α and β electrons by oxidative modification, and finally introducing multi-radical structures into the molecular system and controlling the position of each radical. Herein, successful molecular models with a large magnetic coupling constant (∼3600 cm-1) were obtained. This study paves the way to explore ferromagnetic MGO guided by theoretical study, which may become reality soon.

Perfect matchings in random pentagonal chains

Let G be a (molecule) graph. A perfect matching, or kekule structure and dimer covering, in a graph G is a set of pairwise nonadjacent edges of G that spans the vertices of G. In this paper, we obtained the explicit expression for the expectation of the number of perfect matchings in random pentagonal chains. Our result shows that, for any polygonal chain $$Q_{n}$$ with odd polygons, the number of perfect matchings can be determined by their concatenation LA-sequence.

Some Relations Between Kekule Structure and Morgan-Voyce Polynomials

In this paper, Kekule structures of benzenoid chains are considered. It has been shown that the coefficients of a B_n (x) Morgan-Voyce polynomial equal to the number of k-matchings (m(G,k)) of a path graph which has N=2n+1 points. Furtermore, two relations are obtained between regularly zig-zag nonbranched catacondensed benzenid chains and Morgan-Voyce polynomials and between regularly zig-zag nonbranched catacondensed benzenid chains and their corresponding caterpillar trees.

Biradicaloid with a Twist: Lowering the Singlet–Triplet Gap

Fusing two phenalenyl units such that a Kekule structure is obtained leads to ‘biradicaloid’ systems, which typically display low ­singlet–triplet (S–T) energy gaps. The value of the S–T gap in biradicaloids can be lowered by extending the conjugated system for an additional fused ring(s). Recently, we have described an alternative approach to decrease the S–T gap with the aid of through-space interactions that arise in helically twisted biradicaloids. Using [5]helicene-based biradicaloid ‘cethrene’ as an illustrative example, we have shown that the value of its S–T gap is substantially decreased when compared to that of its planar isomer heptazethrene, on account of the helical twist, inducing crucial intramolecular interactions.

Distributive lattice structure on the set of perfect matchings of carbon nanotubes

Carbon nanotubes are composed of carbon atoms linked in hexagonal shapes, with each carbon atom covalently bonded to three other carbon atoms. Carbon nanotubes have diameters as small as 1 nm and lengths up to several centimeters. Carbon nanotubes can be open-ended or closed-ended (fullerenes). Open-ended single-walled carbon nanotubes are also called tubulenes. The resonance graph R(T) of a tubulene T reflects interactions between Kekule structures—i.e. perfect matchings of T. With the orientation of edges the resonance digraph \(\overrightarrow{R}(T)\) of a tubulene is obtained. As the main result we show that \(\overrightarrow{R}(T)\) is isomorphic to the Hasse diagram of the direct sum of some distributive lattices. Similar results were proved in [10, 16], but one can not directly apply them to tubulenes. As a consequence of the main result it is proved that every connected component of R(T) is a median graph. Further we show that the block graph of every connected component H of the resonance graph of a tubulene is a path and that H contains at most two vertices of degree one.

Perfect matchings in random polyomino chain graphs

Let G be a (molecule) graph. A perfect matching, or Kekule structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekule structures of a graph is interest in chemistry, physics and mathematics. In this paper, we focus on the number of perfect matchings in polyomino chain graphs. Simple exact formulas are given for the expected value of the number of perfect matchings in random polyomino chain graphs and for the asymptotic behavior of this expectation. Moreover, the average value of the number of perfect matchings with respect to the set of all polyomino chain graphs with s square-cells.

Cross-Conjugation at the Heart of Understanding the Electronic Theory of Organic Chemistry

The present status of what has been known and understood about the concept of cross-conjugation is reviewed first and it has been shown how the structure-activity relationship in acyclic and monocyclic conjugated systems can be understood without elaborated formalism but by gathering the knowledge of the basic HMO of linear polyenes. Then it is shown that the mode of electron flow caused by hetero-atom substitution to these hydrocarbons as depicted by the curly arrows on the structural formula is correlated well with the topological structure of cross-conjugated system. Mathematical relations between the conventional HMO and resonance theories especially related to the Kekule structure of conjugated hydrocarbons are explained. The least amount of graph-theoretical techniques, such as the topological index, Z, is also explained. It will be understood how the global trend of the direction and extent of the π-electron flow in acyclic conjugated hydrocarbon networks can be predicted by the diagrammatic method of organic electron theory and how cross-conjugation is an important concept in organic chemistry. Finally several new research trends and observations are introduced and discussed from other than organic chemistry, such as “quantum interference” in physics.

Global Forcing Numbers of Handgun-shaped Benzenoid Systems

Let G be a molecular graph with bond set E(G) that admits a Kekulé structure. A global forcing set of G is any subset of E(G) such that no two Kekulé structures of G coincide on S. The number of bonds in a global forcing set of the smallest cardinality is called the global forcing number of G. The global forcing number gives some sort of identification of the minimal amount of information required to specify Kekule structures of a molecular graph. In this article, we give explicit formulae for global forcing numbers of handgun-shaped benzenoid systems. As a corollary, formulae for the global forcing numbers of parallelogram benzenoid systems can be obtained. [J. Sedlar, The global forcing number of the parallelogram polyhex, Discrete Appl. Math. 160 (2012) 2306-2313]. Keywords: Benzenoid hydrocarbon, global forcing number, global forcing set, handgun-shaped benzenoid system, Kekule structure, perfect matching.

Peierls instability in (5,5) and (9,0) carbon nanotubes: Effect of torsional strain on band gap

Effect of Peierls instability on the electronic band structure of carbon nanotubes (CNTs) under torsional strain is studied by tight-binding approximation. When Kekule structure with two different bond lengths is introduced, armchair (5,5) and zigzag (9,0) CNTs become semiconducting indicating by a small band gap opening at the Fermi level. Numerical calculations show that the increasing of shear strain leads to reduce the band gap of armchair (5,5) carbon nanotube. Shear strain threshold for a semiconductor–metal phase transition depends on the difference in the bond lengths. In the metal state, the torsional strain induces the Fermi wavenumber of (5,5) carbon nanotube deviated from the center of the first Brillouin zone. Under the torsional strain, a semiconductor–metal–semiconductor phase transition appears for (9,0) CNT with Kekule structure as well.

Clar chains and a counterexample

A fullerene is a 3-regular plane graph with only pentagonal and hexagonal faces. The Fries and Clar number of a fullerene are two related parameters, and the Clar number is less understood. We introduce the Clar Structure of a fullerene, a decomposition designed to compute the Clar number for classes of fullerenes. We also settle an open question with a counterexample: we prove that the Clar and Fries number of a fullerene cannot always be obtained with the same Kekule structure.