Benzenoid Graphs Research Articles

Zhang-Zhang Polynomials of Phenylenes and Benzenoid Graphs

The aim of this paper is to study some variations of the Zhang-Zhang polynomial for phenylenes, which can be obtained as special cases of the multivariable Zhang-Zhang polynomial. Firstly, we prove the equality between the first Zhang-Zhang polynomial of a phenylene and the generalized Zhang-Zhang polynomial of some benzenoid graph, which enables us to prove also the equality between the first Zhang-Zhang polynomial and the generalized cube polynomial of the resonance graph. Next, some results on the roots of the second Zhang-Zhang polynomial of phenylenes are provided and another expression for this polynomial is established. Finally, we give structural interpretation for (partial) derivatives of different Zhang-Zhang polynomials.

The Edge-Hosoya Polynomial of Catacondensed Benzenoid Graphs Associated with its Hosoya Polynomial

This paper reveals the relationship between edge-Hosoya polynomial and the Hosoya polynomial of catacondensed benzenoid graphs. The result shows that, for a catacondensed benzenoid graph, the computations of the edge-Hosoya polynomial can be reduced to that of the Hosoya polynomial.

Minimum Number of Kekulé Structures of Certain Benzenoid Graphs and -Polyacenic Nanotubes

In a molecular graph, a Kekulé structure and perfect matching resembles the same. Thus, the number of Kekulé structure(chemically) is equal to the number of perfect matching (mathematically) required to cover the edge set or covalent bonds of a molecular graph entirely. The number of possible Kekulé structures in a molecule (benzenoids) is used to describe the extent of its aromatic property and plays an significant role in the analysis of resonance energy and stability of certain chemical compounds. In this article, the minimum number of Kekulé structures (K) that covers all the edges of a class of peri-condensed benzenoid graphs possessing three rows of hexagon of various length and -polyacenic nanotubes were discussed and summarized.

Degree‐Based Entropy for a Non‐Kekulean Benzenoid Graph

Tessellations of kekulenes and cycloarenes have a lot of potential as nanomolecular belts for trapping and transporting heavy metal ions and chloride ions because they have the best electronic properties and pore sizes. The aromaticity, superaromaticity, chirality, and novel electrical and magnetic properties of a class of cycloarenes known as kekulenes have been the subject of several experimental and theoretical studies. Through topological computations of superaromatic structures with pores, we investigate the entropies and topological characterization of different tessellations of kekulenes. Using topological indices, the biological activity of the underlying structure is linked to its physical properties in (QSPR/QSAR) research. There is a wide range of topological indices accessible, including degree‐based indices, which are used in this work. With the total π‐electron energy, these indices have a lot of iteration. In addition, we use graph entropies to determine the structural information of a non‐Kekulean benzenoid graph. In this article, we study the crystal structure of non‐Kekulean benzenoid graph and then calculate some entropies by using the degree‐based topological indices. We also investigate the relationship between degree‐based topological indices and degree‐based entropies. This relationship is very helpful for chemist to study the physicochemical characterization of non‐Kekulean benzenoid chemical. These numerical values correlate with structural facts and chemical reactivity, biological activities, and physical properties.

On Ve -Degree and Ev -Degree Based Topological Indices for the Series of Benzenoid Graphs

Topological indices play a very important role in the study of structure of a chemical graph. There are many types of topological indices which describe quantitative properties of the structure of different graphs and molecular networks of chemical graphs. In the beginning of the study of topological indices, there were three different types of indices which were based on degree, distance and counting. M. Chellali et al. (“On Ve-Degrees and Ev-Degrees in Graphs,” Discrete Mathematics 340, no. 2 (2017): 31–8) introduced Ve - degree and Ev - degree based topological indices. This article is a contribution to the study of Ve -degree and Ev -degree based topological indices for series of concealed non-Kekulean benzenoid graph. In this article, we have calculated ten different types of topological indices which are based on Ve -degree and Ev -degree for the series of concealed non-Kekulean benzenoid graph.

On the Clar number of graphene fragment

A graphene fragment is a benzenoid graph that its dualist graph is a unicyclic graph. In particular, when the dualist graph of a benzenoid graph is a circle, it is called cyclofusene. In this paper, we determine the Clar number of a cyclofusene graph, and prove a bound for the Clar number of the graphene fragment. Moreover, we construct the graphene fragment which can attain this bound. More precisely, it is shown that the Clar number of the graphene fragment with n hexagons is at most $$[\frac{2n}{3}]$$ .

On Graph theoretic index of Complementary Benzenoids And Macromolecules

Background: A topological index of a graph G is a numerical parameter related to G which characterizes its molecular topology. In the field of QSAR and QSPR research, theoretical properties of the chemical compounds and their molecular topological indices such as distance connectivity indices and degree connectivity indices are used to predict the bioactivity of different molecular compounds. Materials and Methods: Such an approach is different from the traditional QSAR methodology, where one employs selected simpler physico-chemical properties to predict biological activities of molecules. In order to obtain the structure-activity relationships in which theoretical and computational methods are necessary to find appropriate representations of the molecular structure of chemical compounds. These representations are realized through the molecular descriptors. Molecular descriptors are numbers containing structural information derived from the structural representation used for molecules under study. Results: A topological index defined on molecular structure G can be considered as a real valued function f :G→ R+ which maps each durg molecular structure to certain real numbers. Graphene sheets are composed of carbon atoms linked in hexagonal shapes with each carbon atom covalently bonded to three other carbon atoms. Each sheet of graphene is only one atom thick and each graphene sheet is considered a single molecule. Graphene has the same structure of carbon atoms linked in hexagonal shapes to form carbon nanotubes, but graphene is flat rather than cylindrical.. This paper addresses the problem of computing the Wiener , First Zagreb index and Forgotten index of Complementary graphs of graphene sheets, triangular benzenoid graph, circumcoronene molecular graph and nanostar dendrimers. Conclusion: The line graphs were used for modeling amino acid sequences of proteins and of the genetic code. The connected graphs are isomorphic to self complementary graphs. Recently, molecular graphs have proved to be highly useful for drugs activity. Non empirical parameters of chemical structures derived from graph theoretic formalisms are being widely used by many researchers in studies pertaining to molecular design, pharmaceutical drug-design, and environmental hazard assessment of chemicals.

Wiener Index of Edge Thorny Graphs of Catacondensed Benzenoids

The Wiener index is a topological index of a molecular graph, defined as the sum of distances between all pairs of its vertices. Benzenoid graphs include molecular graphs of polycyclic aromatic hydrocarbons. An edge thorny graph G is constructed from a catacondensed benzenoid graph H by attaching new graphs to edges of a perfect matching of H. A formula for the Wiener index of G is derived. The index of the resulting graph does not contain distance characteristics of elements of H and depends on the Wiener index of H and distance properties of the attached graphs.

Resonance graphs of catacondensed even ring systems

A catacondensed even ring system (shortly CERS) is a simple bipartite 2-connected outerplanar graph with all vertices of degree 2 or 3. In this paper, we investigate the resonance graphs (also called Z-transformation graphs) of CERS and firstly show that two even ring chains are evenly homeomorphic iff their resonance graphs are isomorphic. As the main result, we characterize CERS whose resonance graphs are daisy cubes. In this way, we greatly generalize the result known for kinky benzenoid graphs. Finally, some open problems are also presented.

Certain polynomials and related topological indices for the series of benzenoid graphs

A topological index of a molecular structure is a numerical quantity that differentiates between a base molecular structure and its branching pattern and helps in understanding the physical, chemical and biological properties of molecular structures. In this article, we quantify four counting polynomials and their related topological indices for the series of a concealed non-Kekulean benzenoid graph. Moreover, we device a new method to calculate the PI and Sd indices with the help of Theta and Omega polynomials.

Characteristic Graph vs Benzenoid Graph

A characteristic graph is a tree representative of its corresponding benzenoid (cyclic) graph. It may contain necessary information of several properties of benzenoids. The PI-index of benzenoids and their characteristic graphs are compared by correlating it to a structural property (π-electron energy) of the benzenoids using MLR analysis. PI index being applicable to both trees and cyclic graphs, yielded required results for benzenoid and their characteristic graph.

Sd-Index of Another Case of Pericondensed Benzenoid Graphs G(m, n, k)

Sd-Index of Another Case of Pericondensed Benzenoid Graphs G(m, n, k)

Formula for calculating the Wiener polarity index with applications to benzenoid graphs and phenylenes

The Wiener polarity index of a graph is defined as the number of unordered pairs of vertices at distance three. In recent years, this topological index was extensively studied since it has many known applications in chemistry and also in network theory. In this paper, we generalize the result of Behmaram et al. (Appl. Math. Lett. 25:1510–1513, 2012) for calculating the Wiener polarity index of a graph. An important advantage of our generalization is that it can be used for graphs that contain 4-cycles and also for graphs whose different cycles have more than one common edge. In addition, using the main result a closed formula for the Wiener polarity index is derived for phenylenes and recalculated for catacondensed benzenoid graphs. The catacondensed benzenoid graphs and phenylenes attaining the extremal values with respect to the Wiener polarity index are also characterized.

Topological Indices and Their Applications to Circumcised Donut Benzenoid Systems, Kekulenes and Drugs

ABSTRACTThis paper describes a new technique to compute topological indices of donut benzenoids and Kekulenes with applications to drugs by dissecting the original topological network into smaller strength-weighted quotient graphs relative to the transitive closure of the Djokovi–Winkler relation. We have applied this technique to a series of donut benzenoid graphs obtained by circumcising at least two internal hexagons from parent benzenoid graphs. Such donut coronoid graph finds significant applications in emerging chemical materials of importance to synthetic organic chemistry and drug industry. It comprised of donut coronoid structures such as Kekulenes and related circumscribed structures where a number of graph-theoretical based techniques, such as resonance theory and Clar’s sextets. In this work, we have computed a number of topological indices of these donut coronoid systems such as the Wiener index, variants of Szeged, Schultz and Gutman indices.

Analytical expressions for Wiener indices of n-circumscribed peri-condensed benzenoid graphs

We have employed cut axes and vertex partitioning (I-partition) methods to obtain exact analytical expressions for Wiener indices of n-circumscribed peri-condensed benezenoid graphs as a function of n, the order of circumscribing. Such expressions have not been obtained before for nth order circumscribing of peri-condensed benzenoid graphs with the exception of n-Circum-coronene which becomes a honeycomb lattice as n goes to infinity. The Wiener indices of these n-circum peri-condensed benzenoids are found to be polynomials of degree 5. Such expressions can be critical in topological characterization of peri-condensed benzenoid sheets of any order of circumscribing. A number of examples are provided ranging from Circum-polyacenes, Circum-ovalenes, Circum-pyrenes to more complex Circum-peri-condensed graphs of wide ranging complexities.

Topological Indices and Interpolation of Sequences

In this paper, some topological indices of benzenoid graphs and mathematical graphs are computed, using the interpolation of sequences.

On Connectivity Indices of an Infinite Family of the Linear Parallelogram of Benzenoid Graph

A topological index of a graph G is a numeric quantity related to G which is describe molecular graph G. In this paper the Atom Bond Connectivity (ABC) and Geometric-Arithmetic (GA) indices of an infinite class of the linear parallelogram of benzenoid graph.

On Connectivity Indices of an Infinite Family of the Linear Parallelogram of Benzenoid Graph

A topological index of a graph G is a numeric quantity related to G which is describe molecular graph G. In this paper the Atom Bond Connectivity (ABC) and Geometric-Arithmetic (GA) indices of an infinite class of the linear parallelogram of benzenoid graph.

Star-In-Coloring of Benzenoid Graphs And Grid Graphs.

In this paper, we obtained the general pattern of star-in-coloring introduced by Sudha et al.[6] for benzenoid graphs which belong to the series of coronene or circumcoronene graphs and found that its star-in-coloring chromatic number is always 4. We have also obtained the star-in-coloring of grid of squares by considering the cartesian product of two paths and found its chromatic number as 5. We have introduced two new definitions for grid of diamonds and grid of hexagons and found the chromatic number of star-in-coloring of Sudha's grid of complete diamonds and Sudha's grid of complete hexagons to be 5 and 4 respectively. The tensor product of two paths and for all and , in general, with the conditions in our definition give rise to the graph of diamonds with some additional edges. We discussed the star-in-coloring of this graph and found its star-in-coloring chromatic number as 5 for all values of and . Likewise the strong product of two paths and for all and with the conditions in our definition give rise to the graph of hexagons with some additional edges. The star-in-coloring of this type of graphs is also discussed and found its star-in-coloring chromatic number as 4 for all and .

A New Technique to Compute Padmakar-Ivan Index and Szeged Index of Pericondensed Benzenoid Graphs

A New Technique to Compute Padmakar-Ivan Index and Szeged Index of Pericondensed Benzenoid Graphs