Phenylene Chains Research Articles

A Conjugated Phenylene Nanocage with a Guest-Adaptive Deformable Cavity.

The highly strained, phenylene-derived organic cages are typically regarded as very rigid entities, yet their deformation potential and supramolecular properties remain underexplored. Herein, we report a pliable conjugated phenylene nanocage by synergistically merging rigid and flexible building blocks. The anisotropic cage molecule contains branched phenylene chains capped by a calix[6]arene moiety, the delicate conformational changes of which endow the cage with a remarkably deformable cavity. When complexing with fullerene guests, the cage showcases excellent guest-adaptivity, with its cavity volume capable of swelling by as much as 85 %.

A Conjugated Phenylene Nanocage with a Guest‐Adaptive Deformable Cavity

AbstractThe highly strained, phenylene‐derived organic cages are typically regarded as very rigid entities, yet their deformation potential and supramolecular properties remain underexplored. Herein, we report a pliable conjugated phenylene nanocage by synergistically merging rigid and flexible building blocks. The anisotropic cage molecule contains branched phenylene chains capped by a calix[6]arene moiety, the delicate conformational changes of which endow the cage with a remarkably deformable cavity. When complexing with fullerene guests, the cage showcases excellent guest‐adaptivity, with its cavity volume capable of swelling by as much as 85 %.

Alkaline-Stable Poly(phenylene-co-arylene ether) Block Copolymer-Based Anion Exchange Membranes with Alkyl Side Chain

Proton exchange membranes (PEMs) are desirable materials as they can be applied to their potential and wide range of electrochemical applications. However, in most electrochemical applications of PEMs for energy production, precious metals such as platinum are essential as catalysts due to the highly acidic media of the system. Anion exchange membranes (AEMs) have been studied recently to replace these PEMs. However, AEM-based electrochemical devices mainly operate under strongly basic conditions. Under those conditions, AEM is prone to decomposition. As a result, membrane scientists have adopted polymers of ether-free backbone, but their IEC is generally more than 2.0 meq/g to achieve high conductivity. The high IEC results in excessive water uptake, making the AEM fail chemically and mechanically during long-term operations.To develop high-performance AEMs, we have synthesized new phenylene-based block copolymers. They have phenylene chains as hydrophilic segments for high chemical stability, in which hydrophobic parts are composed of arylene ether. This configuration is beneficial in solubility because the hydrophobic arylene ether block mitigates the poor solubility of phenylene. Moreover, a flexible alkyl chain was introduced as conducting moiety into hydrophilic phenylene. Thus, the enhanced solubility promoted the membrane’s efficient hydrophilic/hydrophobic morphological separation. The block copolymer-based AEM showed lower HFR (141 mΩ·cm2) than the Fumatech membrane (168 mΩ·cm2) in single-cell performance at 70 °C.

The Multivariable Zhang–Zhang Polynomial of Phenylenes

The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons.

The minimum Kirchhoff index of phenylene chains

Let G be a connected graph. The resistance distance between any two vertices of G is equal to the effective resistance between them in the corresponding electrical network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index is defined as the sum of resistance distances between all pairs of vertices. Recently, Yang and Wang determined the maximum Kirchhoff index of phenylene chains and proposed a conjecture about the minimum Kirchhoff index. In this note, we characterize the minimum phenylene chains with respect to the Kirchhoff index, thereby prove the conjecture.

Computing the expected subtree number of random hexagonal and phenylene chains based on probability matrices

With structural analysis and probability matrix, this paper studies the subtree number index (number of non-labeled subtrees) of random hexagonal and phenylene chains, and presents exact recursive formulas for the expected values of subtree number index of the random hexagonal and phenylene chains in terms of probability matrices, as applications, we obtain the subtree number of hexagonal linear chain, hexagonal helicence chain, phenylene linear chain and phenylene helicence chain. Moreover, the proposed matrix method also builds a bridge to explore the deep correlation between structural-based and distance-based topological indices, and provides important theoretical support for prediction new properties of chemical compounds.

On expected values of some degree based topological descriptors of random Phenylene chains

Phenylenes belong to a special class of conjugated hydrocarbons composed of a special arrangement of hexagons and squares. In the phenylene structure, every square is adjacent to a pair of hexagons, and any two hexagons are not adjacent. If in the structure of phenylene only two squares are adjacent to every hexagon, then it is called a phenylene chain. In this paper, we compute the expected values of the first and second Gourava index, the first and second redefined Zagreb index, and the Hyper-Zagreb index of random phenylene chains. Furthermore, a comparison between the expected values of the computed indices among the classes of random phenylene chains has been investigated.

Structural characterization and spectral properties of hexagonal phenylene chain network

Structural characterization and spectral properties of hexagonal phenylene chain network

A Method for Computing the Edge-Hosoya Polynomial with Application to Phenylenes

The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance k ≥ 0 in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph G which is obtained by identifying two edges of connected bipartite graphs G1 and G2. To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.

Complete forcing numbers of catacondensed phenylene systems

Combining ?forcing? and ?global? idea, Xu et al. proposed the concepts: complete forcing set and complete forcing number of perfect matchings of graph. In this paper, we give explicit formulae for the complete forcing numbers of phenylene chains and catacondensed phenylene systems, respectively. Moreover, we present an algorithm to find the minimum complete forcing sets of these graphs.

The Expected Values of Hyper Zagreb and Forgotten Topological Indices in Random Phenylene Chains

Hexagonal chains are a special class of data condensed benzenoid system and phenylene chains are a class of polycyclic aromatic compounds. In this paper, we determine the expected values of the Hyper Zagreb and forgotten topological indices for this class of conjugated hydrocarbons. The comparisons between the expected values of these indices concerning the random phenylene chains have been determined explicitly. The graphical illustrations have been given in terms of the differences between the expected values of these indices.

On the Multiplicative Degree-Kirchhoff Indices and the Number of Spanning Trees of Linear Phenylene Chains

Let denote a molecular graph of linear phenylene, containing n hexagons and squares. In this paper, using the decomposition theorem of the normalized Laplacian characteristic polynomial, we obtain that the normalized Laplacian spectrum of consisting of the eigenvalues of two symmetric tridiagonal matrices of order 4n is determined. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above two matrices, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. the number of spanning trees) of is derived.

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

AbstractHexagonal chains are a special class of catacondensed benzenoid system and phenylene chains are a class of polycyclic aromatic compounds. Recently, A family of Sombor indices was introduced by Gutman in the chemical graph theory. It had been examined that these indices may be successfully applied on modeling thermodynamic properties of compounds. In this paper, we study the expected values of the Sombor indices in random hexagonal chains, phenylene chains, and consider the Sombor indices of some chemical graphs such as graphene, coronoid systems and carbon nanocones.

The Gutman Index and Schultz Index in the Random Phenylene Chains

The Gutman index and Schultz index are two topological indices‎. ‎In this paper‎, ‎we first give exact formulae for the expected values of the Gutman index and Schultz index of random phenylene chains‎, ‎and we will also get the average values of the Gutman index and Schultz index in phenylene chains.‎

Extremal phenylene chains with respect to the Mostar index

For a connected graph [Formula: see text], the Mostar index is defined as [Formula: see text], where [Formula: see text] (respectively, [Formula: see text]) is the number of vertices of [Formula: see text] closer to [Formula: see text] (respectively, [Formula: see text]) than [Formula: see text] (respectively, [Formula: see text]). In this paper, we determine the first five maximal (respectively, the first five minimal) values of the Mostar index among all phenylene chains with [Formula: see text] hexagons, the corresponding extremal chains are completely characterized, respectively.

Extremal phenylene chains with respect to detour indices

Computing topological indices of molecular structures is a fundamental and classical topic. In organic chemistry, hexagonal and quadrilateral molecular structures are very common. The detour index $$\omega (G)$$ of a connected graph G is defined as $$\omega (G)=\sum \nolimits _{\{u,v\}\subseteq V(G)}l_{G}(u,v)$$ , where $$l_{G}(u,v)$$ denotes the detour distance between vertices u and v. In this study, we obtain the explicit analytical expression for detour index of phenylene chains with a fixed number of hexagons. Further minimal and maximal phenylene chains are obtained.

The expected values of some indices in random phenylene chains

A special class of conjugated hydrocarbons is known as phenylenes, which is composed of a special arrangement of six- and four-membered rings. In particular, any two six-membered rings (hexagons) are not adjacent, and every four-membered ring (square) is adjacent to a pair of nonadjacent hexagons. If each hexagon of phenylene is adjacent only to two squares, then the obtained chain is called the phenylene chain. The main object of this paper is to determine the expected values of the sum-connectivity, harmonic, symmetric division, variable inverse sum degree and general Randic indices for this class of conjugated hydrocarbons. The comparisons between the expected values of these indices with respect to the random phenylene chains have been determined explicitly. The graphical illustrations have been given in terms of the differences between the expected values of these indices.

The Expected Values of Merrifield-Simmons Index in Random Phenylene Chains

The Merrifield-Simmons index of a graph G is the number of independent sets in G. In this paper, we give exact formulae for the expected value of the Merrifield-Simmons index of random phenylene chains by means of auxiliary graphs.

The expected values of arithmetic bond connectivity and geometric indices in random phenylene chains

Phenylenes are a class of conjugated hydrocarbons composed of a special arrangement of six- and four-membered rings. In this paper, the expected values of the atom-bond connectivity (ABC), geometric-arithmetic (GA), and Zagreb indices of this class of conjugated hydrocarbons have been determined. At the end, the comparisons with respect to the random phenylene chains among the expected values of these indices, have been determined explicitly. The graphical profiles of these indices have been shown in order to support our results.

On the Kirchhoff Indices and the Number of Spanning Trees of Möbius Phenylenes Chain and Cylinder Phenylenes Chain

Let denote a molecular graph of linear phenylene. Then the Möbius phenylenes chain is the graph obtained from the by identifying the opposite lateral edges in reversed way, whereas the cylinder phenylene chain is the graph obtained from the by identifying the opposite lateral edges in ordered way. In this paper, by the decomposition theorem of Laplacian polynomial and the relationship between the roots and the coefficients of the characteristic polynomial, explicit closed-form formulae of the Kirchhoff index (resp. the number of spanning trees) of and are obtained.