The simplest coronoids: Hollow hexagons

Hollow hexagons form a subclass of primitive coronoid systems. The macrocyclic aromatic hydrocarbon kekulene corresponds to a hollow hexagon. The hollow hexagons for given numbers of hexagonal units (h) were enumerated by computer aid, but also an analytical solution for the numbers of hollow hexagons was achieved. For the Kekulé structure counts (k) of a hollow hexagon a general formula is reported. Also the maximum and minimumK numbers are considered.This article is Part VI of the series Enumeration and Classification of Coronoid Hydrocarbons. For Part V, see ref. [36].     

<Emphasis Type="Italic">k</Emphasis>-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 Kekulé 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. *This work is supported by FRG, Hong Kong Baptist University; NSFC and TRAPOYT.     

Cyclical Edge-Connectivity of Fullerene Graphs and (k, 6)-Cages

It is shown that every fullerene graph G is cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. The result is then generalized to the case of (k,6)-cages, i.e., polyhedral cubic graphs whose faces are only k-gons and hexagons. Certain linear and exponential lower bounds on the number of perfect matchings in such graphs are also established.     

Kekulé structure counts in coronoid hydrocarbons: A general solution

The Kekulé structure count (K) of a planar graph (in the graph-theoretical sense) is obtainable as the Pfaffian of a skew-symmetric adjacency matrix. This principle was used to design a general computer program for theK numbers of coronoid systems. Thus it became possible for the first time to determineK, as a matter of routine, during the computer-aided generation and enumeration of all types of Kekuléan coronoids, including those where perimeters of [4k] cycles are present. The methods are exemplified by a complete account (with figures andK numbers] of the 27 half essentially disconnected coronoids with the phenalene hole and 11 or 12 hexagons.     

Hexagonal resonance of (3,6)-fullerenes

A (3,6)-fullerene G is a plane cubic graph whose faces are only triangles and hexagons. It follows from Euler’s formula that the number of triangles is four. A face of G is called resonant if its boundary is an alternating cycle with respect to some perfect matching of G. In this paper, we show that every hexagon of a (3,6)-fullerene G with connectivity 3 is resonant except for one graph, and there exist a pair of disjoint hexagons in G that are not mutually resonant except for two trivial graphs without disjoint hexagons. For any (3,6)-fullerene with connectivity 2, we show that it is composed of n(n ≥ 1) concentric layers of hexagons, capped on each end by a cap formed by two adjacent triangles, and none of its hexagons is resonant.     

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(k3)-resonant tubules. The lower bound of its Clar number of k(k3)-resonant tubules is also given. Note that the present paper does not consider the capped species.     

Graphical properties of polyhexes: Perfect matching vector and forcing

From the viewpoint of graph theory and its applications, subgraphs of the tiling of the plane with unit squares have long been studied in statistical mechanics, In organic chemistry, a much more relevant case concerns subgraphs of the tiling with unit hexagons. Our purpose here is to take a mathematical view of such polyhex graphsG and study two novel concepts concerning perfect matchingsM. First, the forcing number ofM is the smallest number of edges ofM which are not contained in any other perfect matching ofG. Second, the perfect matching vector ofM is written (n ₃,n ₂,n ₁,n ₀), wheren _k is the number of hexagons with exactlyk edges inM. We establish some initial results involving these two concepts and pose some questions.     

Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

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.     

On the Ordering of a Class of Graphs with Respect to their Matching Numbers

Based on the number of k-matching m(G,k) of a graph G, Gutman and Zhang introduced an order relation of graphs: for graphs G ₁ and G ₂, if m(G ₁,k) ≥ m(G ₂,k) for all k. The order relation has important applications in comparing Hosaya indices and energies of molecular graphs and has been widely studied. Especially, Gutman and Zhang gave complete orders of six classes of graphs with respect to the relation . In this paper, we consider a class of graphs with special structure, which is a generalization of a class of graphs studied by Gutman and Zhang. Some order relations in the class of graphs are given, and the maximum and minimum elements with respect to the order relation are determined. The new results can be applied to order some classes of graphs by their matching numbers.     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).      

Maximal resonance of cubic bipartite polyhedral graphs

Let H{\mathcal{H}} be a set of disjoint faces of a cubic bipartite polyhedral graph G. If G has a perfect matching M such that the boundary of each face of H{\mathcal{H}} is an M-alternating cycle (or in other words, G-H{G-\mathcal{H}} has a perfect matching), then H{\mathcal{H}} is called a resonant pattern of G. Furthermore, G is k-resonant if every i (1 \leqslant i \leqslant k){i\,(1\,\leqslant\, i\, \leqslant\, k)} disjoint faces of G form a resonant pattern. In particular, G is called maximally resonant if G is k-resonant for all integers k \geqslant 1{k\,\geqslant\, 1}. In this paper, all the cubic bipartite polyhedral graphs, which are maximally resonant, are characterized. As a corollary, it is shown that if a cubic bipartite polyhedral graph is 3-resonant then it must be maximally resonant. However, 2-resonant ones need not to be maximally resonant.     

Steiner degree distance indices: Chemical applicability and bounds

The k-center Steiner degree distance ( SDD _k(G) ) has recently been introduced as a natural extension of the degree distance DD(G). In this paper, the prediction potential of SDD _k(G) is discussed. Then, the relation between this and some other well-known distance-based indices of trees is derived to explain its prediction potential. Finally, the lower and upper bounds of SDD _k(G) in terms of some other graph invariants are presented.     

On the existence of multiply connected monolayered cyclofusenes with given parameters

For a given bipartite graph G its skewness is defined as the difference between the sizes of its classes of bipartition. We show that a multiply connected monolayered cyclofusene with m ≥ 2 internal holes and skewness k exists if and only if 0 ≤ k ≤ 2m − 2, thus settling in affirmative two conjectures raised in a recent paper by Karimi et al.     

The Effect of Anomeric Head Groups, Surfactant Hydrophilicity, and Electrolytes onn-Alkyl Monoglucoside Microemulsions

The effects of the variables of head group structure and salt concentration on microemulsions formed in mixtures of water, alkyl ethylene glycol ethers (C_kOC₂OC_k), andn-alkyl β- -glucopyranosides (C_mβG₁) are explored. Phase behavior of mixtures containing an anomer of the surfactant (n-alkyl α- -glucopyranoside, C_mαG₁), or surfactants with long head groups (n-alkyl maltopyranosides, C_mG₂), or NaCl or NaClO₄as electrolyte are systematically reported as a function of temperature and composition. The substitution ofn-alkyl α- -glucopyranosides forn-alkyl β- -glucopyranosides causes precipitation under some conditions in all mixtures studied. These solubility boundaries begin in the water–surfactant binary mixture at the Krafft boundary, then extend to high concentrations of both surfactant and oil. Increasing the effective length of the surfactant head group by adding C_mG₂to water–C_kOC₂OC_k–C_mβG₁mixtures moves the phase behavior dramatically up in temperature when even small amounts of C_mG₂are used. Adding a lyotropic electrolyte, NaCl, to water–C_kOC₂OC_k–C_mβG₁mixtures moves the phase behavior down in temperature, while the hydrotropic electrolyte NaClO₄moves the phase behavior up in temperature.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

Nitrous Oxide as Carrier Gas in Capillary Gas-Liquid Chromatography

On use of nitrous oxide as carrier gas the retention factors of the chromatographed compounds decrease linearly with increasing average column pressure. Other retention characteristics (relative retention, retention index) change linearly. This effect was demonstrated by using a capillary column coated with nonpolar polydimethylsiloxane phase SE-30. As shown for capillary GLC the linear correlation is valid for the same column:k_i(G₁,P₁) = A k_i(G₂, P₂) + B, where k_i(G₁, P₁) and k_i(G₂,P₂) are the retention factors of compound i at average column pressures P₁ and P₂ when using carrier gases G₁ and G₂, respectively; A and B are coefficients.     

The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons

The connective eccentricity index (CEI) of a graph G is defined as , where ε_G(.) denotes the eccentricity of the corresponding vertex. The CEI obligates an influential ability, which is due to its estimating pharmaceutical properties. In this paper, we first characterize the extremal graphs with respect to the CEI among k-connected graphs (k-connected bipartite graphs) with a given diameter. Then, the sharp upper bound on the CEI of graphs with given connectivity and minimum degree (independence number) is determined. Finally, we calculate the CEI of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons. We compare their CEI with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the CEI is better than or as good as the models corresponding to the other distance-based indices.     

Driven translocation of semiflexible polyelectrolyte through a nanopore

The influence of the chain stiffness on the translocation of semiflexible polyelectrolyte through a nanopore is investigated using Langevin dynamics simulations. Results show that the translocation time τ increases with the bending modulus k_θ because of the increase of viscous drag forces with k_θ. We find that the relation between τ and k_θ, the asymptotic behavior of τ on the polyelectrolyte length N, and the scaling relation between τ and the driving force f are dependent on k_θ and N. Our simulation results show that the semiflexible polyelectrolyte chain can be regarded as either a flexible polyelectrolyte at small k_θ or large N where its radius of gyration R_G is larger than the persistence length L_p or a stiff polyelectrolyte at large k_θ or short N where R_G < L_p. Results also show that the out‐of‐equilibrium effect during the translocation becomes weak with increasing k_θ. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 912–921     

Atomic-scale investigation of the interaction between coniferyl alcohol and laccase for lignin degradation using molecular dynamics simulations and spectroscopy

The interaction between coniferyl alcohol (CA) and laccase (LAC) was investigated using molecular dynamics (MD) simulations and spectral experiments. The mode of interaction between CA and LAC was established by MD simulations. The micro-environmental changes, stability and rigidity of the LAC-CA system were assessed by relevant parameters. These parameters include root mean square deviation (RMSD), root mean square fluctuation (RMSF) and radius of gyration (Rg). The calculated binding free energy (ΔG_binding=??19.99?kcal·mol.^?1), the van der waals (VDW) contribution (ΔG_vdw=?23.99?kcal·mol^?1) and the electrostatic energy (ΔG_ele=?23.09?kcal·mol^?1) of LAC-CA system demonstrated that the interaction of LAC-CA was a spontaneous process and the main interaction forces were van der Waal's and electrostatic forces. The values of ΔG_vdw and ΔG_ele were negative, which demonstrated that VDW interactions and electrostatic interactions were favorable for the binding of CA and LAC. The binding constants, thermodynamic parameters, molecular force types and binding distances confirmed the interaction between CA and LAC and further verified the rationality of the theoretical model by spectral experiments. The MD simulations and experimental approaches provide clues for the discovery of new mediators and useful references for the mechanism of microbial degradation of lignin and industrialization of lignocellulose.