Benzenoid Systems Research Articles

Enumeration of the Hosoya index of pericondensed benzenoid system

In chemical graph theory, topological indices play an important role and have various uses in quantitative structure-property relationships (QSPR) as well as quantitative structure-activity relationships (QSARs). The Hosoya index is one of them, performing a molecular descriptor in mathematical chemistry. Therefore, progressing with investigations requires computing the Hosoya index of distinct molecular graphs. This research presents a computational method for determining the Hosoya index of pericondensed benzenoid systems using the transfer matrix approach and the Hosoya vector.

Group Theoretic Approach towards the Balaban Index of Catacondensed Benzenoid Systems and Linear Chain of Anthracene

In this work, we present the analytical closed forms of the Balaban index for anthracene and catacondensed benzenoid systems using group theoretic techniques. The Balaban index is a distance-based topological index that provides valuable information about the properties of chemical structures. We emphasize the importance of determining analytical closed forms of the Balaban index for catacondensed benzenoid systems and linear chains of anthracene, as it enables a deeper understanding of these systems and their behavior. Our analysis utilizes the group action of the automorphism group of these chains on the set of vertices, which refer to the points where the chains intersect. In future work, we plan to determine the Balaban index of other polymeric linear chains using group theoretic techniques and extend the applications of this index to other fields, such as materials science and biology. It is clear that the Balaban index remains a valuable tool in theoretical and computational chemistry, and its applications are constantly evolving.

Computing Topological Indices of Certain Networks

In mathematical chemistry, topological indices are molecular descriptors that are calculated on the molecular graph of a chemical compound. The molecular graph is a graph which is obtained from some chemical structures. The degree of every molecular graph cannot exceeds 4. Topological indices are numerical quantities of a graph that describe its topology. An atom represents a vertex and a bond between two atoms represents an edge in a molecular graph. Mainly there are three types of topological indices viz., degree-based, distance based and eigenvalue-based topological indices. The first degree-based topological indices are the first and second Zagreb indices. The first Zagreb index M_1 is defined as the sum of squares of degrees of each vertex in a graph G and the second Zagreb index M_2 is the product of degree of every adjacent vertices. In this case the summation goes on the set of edges of a graph G. The most studied topological indices are degree-based topological indices. Motivated by these topological indices in this paper, we introduce five new degree-based topological indices based on the neighborhood degree of a vertex. Further, we compute the values of various nanostructures like hexagonal parallelogram P(m,n) nanotube, triangular benzenoid G_n,zigzag-edge coronoid fused with starphene nanotubes ZCS(k,l,m), dominating derived networks D_1,D_2,D_3, Porphyrin Dendrimer, Zinc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimer, Poly(Ethylene amido amine Dendrimer, PAMAM dendrimers(????????1,????????2,????????1), linear polyomino chain L_n,Z_n,B_n^1 (n≥3),B_n^2 (n≥3) and triangular, hourglass, and jagged-rectangle benzenoid systems of these indices. The standard computational techniques are used for the computation of topological indices of nanostructures. For the edge partition of the nanostructures the algebraic techniques are used. Using these techniques computation of topological indices became easy and also helped to get the more accurate results.

Sombor index and elliptic Sombor index of benzenoid systems

Let G be a graph with vertex set V and edge set E. A topological index has the formTI(G)=∑uv∈Ef(du,dv), where f=f(x,y) is a pertinently chosen function which must be symmetric and real-valued for all x,y pertaining to vertex degrees of the graph G. Particularly interesting are the Sombor index SO and the elliptic Sombor index ESO, induced by the functions f(x,y)=x2+y2 and f(x,y)=(x+y)x2+y2, respectively. In this paper we analyze the ordering relations in benzenoid systems with respect to these two important topological indices. Also, we extend the results to general Sombor index SOα,β and general elliptic Sombor index ESOα.

Structure of [18]Annulene Revisited: Challenges for Computing Benzenoid Systems.

For cyclic conjugated structures, erratic computational results have been obtained with Hartree-Fock (HF) molecular orbital (MO) methods as well as density functional theory (DFT) with large HF-exchange contributions. In this work, the reasons for this unreliability are explored. Extensive computations on [18]annulene and related compounds highlight the pitfalls to be avoided and the due diligence required for such computational investigations. In particular, a careful examination of the MO singlet-stability eigenvalues is recommended. The appearance of negative eigenvalues is not (necessarily) problematic, but near-zero (positive or negative) eigenvalues can lead to dramatic errors in vibrational frequencies and related properties. DFT approaches with a lower HF admixture generally appear more robust in this regard for the description of benzenoid structures, although they may exaggerate the tendency toward planarity and C-C bond-equalization. For the iconic [18]annulene, the results support a nonplanar equilibrium structure. The density-fitted frozen natural orbital coupled-cluster singles and doubles with perturbative triples [DF-FNO CCSD(T)] method of electron correlation with an aug-pVQZ/aug-pVTZ basis set places the C2 global minimum 1.1 kcal mol-1 below the D6h stationary point.

MODIFIED ELLIPTIC SOMBOR INDEX AND ITS EXPONENTIAL OF A GRAPH

The elliptic Sombor index is a recently introduced topological index. In this paper, we put forward the modified elliptic Somborindex and its corresponding exponential of a graph and compute exact formulas for some families of benzenoid systemssuch as triangular benzenoids, benzenoid rhombus, benzenoid hourglass and jagged rectangle benzenoid systems

REDUCED ELLIPTIC AND MODIFIED REDUCED ELLIPTIC INDICES OF SOME CHEMICAL STRUCTURES

In this paper, we introduce the reduced elliptic index, the modified reduced elliptic index and their corresponding exponentials of a graph. Also we determine these newly defined elliptic indices for some standard graphs, jagged-rectangle benzenoid systems and polycyclic aromatic hydrocarbons which are appeared in Mathematical Chemistry

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.

Entropy Measures of Topological Indices Based Molecular Structure of Benzenoid Systems

Graph theory plays a significant role in the applications of chemistry, pharmacy, communication, maps, and aeronautical fields. A benzenoid is a class of chemical compounds with at least one benzene ring(hexagon as a graph) and resonance bonds in the benzene ring give increased stability in benzenoids. The molecules of chemical compounds are modeled as a graph to study the properties of the compounds. The geometric structure of the compound relates to a few physical properties such as boiling point, enthalpy, π-electron energy, and molecular weight. Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of configurations that molecules might take. Degree-based entropy is used to address a wide range of problems in the domains of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines. This paper focusses on computing analytical expressions of degree-based entropy measures for benzenoid systems.

Topological Properties of some Benzenoid Chemical Structures by Using Face Index

A variety of graphical invariants have been described and tested, offering lots of applications in the fields of Nano chemistry, computational networks and different scientific research areas. One commonly studied group of invariants is the topological index, which allows the prediction of the chemical, biological and physical properties of a chemical structure. Topological indexes are numerical quantities that can be used to describe the properties of the molecular graph. In this article, we computed the face index formula for certain molecular structures of benzenoid series such as polycyclic aromatic hydrocarbon (PAH), jagged-rectangle benzenoid system (JRB[m,n]) and the series of concealed non-kekulean benzenoid system.

Wiener Index Calculation on the Benzenoid System: A Review Article

The Weiner index is considered one of the basic descriptors of fixed interconnection networks because it provides the average distance between any two nodes of the network. Many methods have been used by researchers to calculate the value of the Wiener index. starting from the brute force method to the invention of an algorithm to calculate the Wiener index without calculating the distance matrix. The application of the Wiener index is found in the molecular structure of organic compounds, especially the benzenoid system. The value of the Wiener index of a molecule is closely related to its physical and chemical properties. This paper will show a comprehensive bibliometric survey of peer-reviewed articles referring to the Wiener index of benzenoid. The Wiener index values of several benzenoid compounds using cubic polynomial are also reported. The Wiener index of benzenoid supports much of the research and provides productive citations for citing the study. Keywords: Wiener index, benzenoid, distance matrix, chemical properties, cubic polynomial, topological.

Wiener Index Calculation on the Benzenoid System: A Review Article

The Weiner index is considered one of the basic descriptors of fixed interconnection networks because it provides the average distance between any two nodes of the network. Many methods have been used by researchers to calculate the value of the Wiener index. starting from the brute force method to the invention of an algorithm to calculate the Wiener index without calculating the distance matrix. The application of the Wiener index is found in the molecular structure of organic compounds, especially the benzenoid system. The value of the Wiener index of a molecule is closely related to its physical and chemical properties. This paper will show a comprehensive bibliometric survey of peer-reviewed articles referring to the Wiener index of benzenoid. The Wiener index values of several benzenoid compounds using cubic polynomial are also reported. The Wiener index of benzenoid supports much of the research and provides productive citations for citing the study. Keywords: Wiener index, benzenoid, distance matrix, chemical properties, cubic polynomial, topological.

Partition of topological indices of benzenoid hydrocarbons into ring contributions

AbstractThis work presents a simple method for partitioning the bond‐additive and atoms‐pair‐additive distance‐based topological indices of plane graphs into the sum of contributions of inner faces. The proposed method is applied to decompose several topological indices (Wiener, hyper‐Wiener, Tratch‐Stankevich‐Zefirov, Balaban, and Szeged indices) into the ring contributions for a series of benzenoid systems. It was found that the employed ring partitioning scheme is providing an accurate assessment of the dominant cyclic conjugation modes in the studied benzenoid hydrocarbons. Thus, the proposed method can be used as the alternative for the quantum‐chemistry‐based aromaticity indices which are significantly more computationally demanding.

Sum Augmented and Multiplicative Sum Augmented Indices of Some Nanostructures

We put forward the sum augmented index, multiplicative sum augmented index of a graph. We determine the sum augmented index and the multiplicative sum augmented index for polycyclic aromatic hydrocarbons and jagged rectangle benzenoid systems.

Multiplicative Revan-Sombor Indices of Some Benzenoid Systems

In this paper, we introduce the multiplicative modified Revan Sombor index of a graph. Also we compute the multiplicative Revan Sombor index, multiplicative modified Revan Sombor index of triangular benzenoids, benzenoid rhombus, benzenoid hourglass and benzenoid systems.

Vertex Degree Based Analysis for Donut Benzenoid System

The donut benzenoid system is a subclass of polycyclic aromatic hydrocarbons (PAHs) that is produced when a few hexagons are selectively removed from planar benzenoid networks. The nature and behavior of the donut benzenoid system are determined by the cavity that is generated as a result. The donut benzenoid system that has been synthesized up to this point has several geometric and electrical features that are entirely unique. By adjusting the dimensions and contours of the cavity, it is possible to modify these properties so that they are better suited to a given application. Therefore, doing theoretical research on this system from a structural point of view can be of great assistance in the process of designing and synthesizing a new form of donut benzenoid system. This article is primarily concerned with the process of obtaining the analytical formulations for four different types of Vertex Degree Based (VDB) indices, including Multiplicative Topological Indices (TIs), M-Polynomial, Entropy Measures, and Irregularity based TIs for the donut benzenoid system.

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.

A Method for Computing the Merrifield–Simmons Index on Benzenoid Systems

We present a method for computing the Merrifield-Simmons index on some basic graphs. For example, our proposal works on paths, simple cycles, trees, and we show that the method can be extended to process benzenoid systems Hr,t with a total of r.t hexagons. In the case of benzenoid systems, our method consists of performing a linear-time Hamiltonian walking on an isomorphic hexagonal grid from , at the same time that the number of independent sets are being computed incrementally. Our method improves the asymptotic behavior of the time-complexity with respect to the transfer matrix method, which is the classic method for computing the Merrifield-Simmons index on grid graphs, even for benzenoid systems.

Computing the Number of Matchings in Catacondensed Benzenoid Systems

The Hosoya index of is defined as the total number of independent edge sets (number of -matchings ) in . The Hosoya index is one of the most important topological indices in the field of mathematical chemistry because of its relationship with several thermodynamic properties. Therefore, computation of the number of -matchings of various molecular structures has importance. Two methods, one for computing the number of the Hosoya index of catacondensed benzenoid systems and the other for the number of -matchings in benzenoid chains (unbranched catacondensed benzenoid systems), have been presented so far. In this paper, a method based on some transfer matrices to compute the number of -matchings of arbitrary (both unbranched and branched) catacondensed benzenoid systems is presented. Moreover, some algorithms are designed to keep the applicability of the method the same as increases.

Computation method of the Hosoya index of primitive coronoid systems.

Coronoid systems are natural graph representations of coronoid hydrocarbons associated with benzenoid systems, but they differ in that they contain a hole. The Hosoya index of a graph G is defined as the total number of independent edge sets, that are called k-matchings in G. The Hosoya index is a significant molecular descriptor that has an important position in QSAR and QSPR studies. Therefore, the computation of the Hosoya index of various molecular graphs is needed for making progress on investigations. In this paper, a method based on the transfer matrix technique and the Hosoya vector for computing the Hosoya index of arbitrary primitive coronoid systems is presented. Moreover, the presented method is customized for hollow hexagons by using six parameters. As a result, the Hosoya indices of both each arbitrary primitive coronoid system and also each hollow hexagon can be computed by means of a summation of four selected multiplications consisting of presented transfer matrices and two vectors.