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Graph theory plays a crucial role in various applications of mathematics and applied sciences. One specialised branch of graph theory is mathematical chemistry, which focuses on mathematical modelling and analysing chemical compounds and their properties. In this context, graphs are used to represent the structural and topological features of molecules, enabling chemists to gain insights into chemical reactions and make predictions about molecular properties. Recently, new versions of Sombor indices have been introduced using a geometric approach. This article specifically focuses on entropy-based variations of these Sombor indices, which includes SO, $ SO_{red} $ S O red , $ SO_{avg} $ S O avg , $ {}^mSO $ m SO , $ SO_3 $ S O 3 and $ SO_4 $ S O 4 , in the context of graphene sheet. Graphene has gained significant attention in the scientific and technological communities due to its exceptional properties. It finds widespread applications in diverse fields such as nanotechnology, electronics, energy storage, sensors, materials science and optoelectronics. Given the promising applications of graphene, it becomes essential to theoretically analyse its structure. Molecular descriptors play a crucial role, as they are strongly linked to various characteristics of chemical compounds. To better understand the Sombor indices, this article graphically represents their entropy measures. [ABSTRACT FROM AUTHOR]

Topological descriptors have drawn more interest recently because of the simplicity of generation and quick evaluation times, by avoiding time-consuming laboratory experiments. The distance based descriptors stand out as being pivotal in the QSAR and QSPR analysis of physico-chemical aspects. Dendrimers are branching polymeric molecules distinguished by its systematic growth pattern. Non-conjugated Ethylene oxide dendritic core attached with tetraphenylethylene is one such hyper branched polymer with strong solubility in a majority of organic solvents making them an active constituent for a variety of experimental purposes. No significant amount of research has been carried over based on the topological aspects of this dendrimer. This study effectively establishes the correlation between the chemical and physical attributes of the structure which can further aid in experimental research. In this work, various fundamental distance based descriptors of Ethylene oxide cored dendrimer have been evaluated by converting the original graph to quotient graphs using Djokovic̀-Winkler relation. Moreover, a graphical comparison of these descriptors has also been plotted to help comprehend the growth trend of these numerical values. [ABSTRACT FROM AUTHOR]

Macromolecules are gaining much attention in various fields today. Dendrimers are artificially synthesized macromolecules by convergent or divergent approach. They are compact regular structures with spherical dimension and has a vast number of application in disparate fields such as drug delivery, material science, and biology, magnetic resonance imaging, an organic light-emitting device, etc. Determining the pharmacological, chemical, and biological characteristics of a substance necessitates a significant amount of effort. From the chemical graph of the dendrimer structure, we can infer those characteristics with the help of numerical descriptors known as the topological index. The Wiener and Szeged indices are two important distance-based topological indices applicable in nanoscience. The degree-based topological indices also have great importance and huge applications in structural chemistry. These indices together with graph entropy are found to be more effective and have found application in different sciences. In this work, the Wiener index, Szeged indices, Mostar indices, and Padmakar Ivan index for cyclen cored dendrimers are evaluated by converting the original graph into quotient graphs using Θ * - classes. This technique is applied since the regular cut method is a lengthy process while moving on to higher generations and also due to the presence of odd cycles in the structure. The degree-based indices and the degree-based graph entropies for the cyclen cored dendrimers are further studied. The comparison graphs with respect to the topological indices as well as graph entropies are also presented. [ABSTRACT FROM AUTHOR]

Nanosheets with boron elements have excellent characteristics which makes the boron polymorphs unique and super hard. A boron α -icosahedral nanosheet in crystalline form has superconductivity and thermal electronic properties. In theoretical chemistry and QSPR/QSAR study, a topological descriptor is an important analytical tool. It helps to analyse the structure and its properties and also correlates the with numerical expressions. The valence-based M-polynomial provides quantitative measures of molecular properties based on their geometric, electrostatic, and quantum chemical characteristics. In this article, the QSPR/QSAR analysis is performed for this nanosheet and the analytical expressions are validated with original synthesized data, and received excellent correlation values of 0.9835 and 0.9932. The mathematical expression of the structure is analysed and the indices are compared graphically and numerically. [ABSTRACT FROM AUTHOR]

Chemical and biological properties of polycyclic aromatic compounds rely intimately upon their structures such as the fundamental molecular topology. In this way, quantitative structure-activity and property relationship (QSAR/QSPR) strategies have been formulated for predicting properties of polycyclic benzenoid hydrocarbons and related graphs from their chemical structures. In this paper, we give the exact analytical expression of three types of polycyclic aromatic compounds where the base molecule of these structure contains hexabenzocorenene (HBC). Furthermore, we research the results by MATLAB and obtain the relationship of the indices which they portray the physcio-chemical properties. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce two new classes of series type additive functional Equations (FEs). The first class of equations is derived from the sum of the squares of the alternative series and the second one is obtained from the sum of the cubes of the series. The solution of the FE is investigated using the principle of mathematical induction. The beauty of this method lies in the fact that it satisfies the property of the additive FE as well as the series. Banach spaces are one of the widely-used spaces that are very helpful to analyse the stability results of various FEs. The Banach space conditions have been applied and the stability results are established for both of the equations. Furthermore, the Banach Contraction principle and alternative of fixed point theorem are used to derive the stability results in a fixed point technique (FPT). The relationship between the FEs and both the series is established through the principle of mathematical induction in the Application section, which adds novelty to the derived results. [ABSTRACT FROM AUTHOR]

In this article, a new kind of bilateral symmetric additive type functional equation is introduced. One of the interesting characteristics of the equation is the fact that it is ideal for investigating the Ulam–Hyers stabilities in two prominent normed spaces, namely fuzzy and random normed spaces simultaneously. This article analyzes the proposed equation in both spaces. The solution of this equation exhibits the property of symmetry, that is, the left of the object becomes the right of the image, and vice versa. Additionally, the stability results of this functional equation are determined in fuzzy and random normed spaces using direct and fixed point methods. [ABSTRACT FROM AUTHOR]

Dendrimers are well-defined nanoparticles, which have far-reaching application in the field of chemistry. Many efforts have been devoted to development of dendrimer due to thier unique structure and various properties and broad application. It helps in varieties of purposes as a catalyst in drug delivery and drug design. The topological descriptor analyze the structure–property relationship of chemical compounds. In this paper, some numerical expressions have been obtained to understand the behavioral pattern of chiral polyamidoamine (PAMAM) dendrimer and PAMAM anthracene moieties dendrimer. The analytical expression has been plotted and compared with varieties of indices to show how it varies between each indices. [ABSTRACT FROM AUTHOR]

In this paper, a novel coronavirus SIDARTHE epidemic model system is constructed using a Caputo-type fuzzy fractional differential equation. Applying Caputo derivatives to our model is motivated by the need to more thoroughly examine the dynamics of the model. Here, the fuzzy concept is applied to the SIDARTHE epidemic model for finding the transmission of the coronavirus in an easier way. The existence of a unique solution is examined using fixed point theory for the given fractional SIDARTHE epidemic model. The dynamic behaviour of COVID-19 is understood by applying the numerical results along with a combination of fuzzy Laplace and Adomian decomposition transform. Hence, an efficient method to solve a fuzzy fractional differential equation using Laplace transforms and their inverses using the Caputo sense derivative is developed, which can make the problem easier to solve numerically. Numerical calculations are performed by considering different parameter values. [ABSTRACT FROM AUTHOR]

Topological indices (TIs) are numerical quantities that enable theoretical chemists to analyse molecular structures mathematically. These TIs are essential to exploring chemical compounds using theoretical techniques like QSAR/QSPR methods. Superphenalene is a large polycyclic aromatic hydrocarbon molecule which has been quickly gaining importance as a building block for alternate energy providers due to its photovoltaic properties. The exciting features of this compound, coupled with its potential applications, warrant an investigation of its nature and properties from a structural perspective. The objective of this research is to compute the proper analytical expressions of four kinds of vertex degree-based (VDB) indices for superphenalene. The numerical values of these indices and 3D graphical representations also help in understanding the relationship between the VDB indices of the compound and its underlying chemical structure quantitatively. [ABSTRACT FROM AUTHOR]

Benzenoid systems are a notable class of polycyclic aromatic compounds which have applications ranging from household materials to advanced fields like nanotechnology. Hence, the study of their fundamental properties, like observed bioactivities, carcinogenicity, toxicity, and other significant characteristics, have fascinated theoretical chemists for quite a long time. From this perspective, the topological descriptors of these networks, facilitate the prediction of their molecular properties through quantitative structure-activity (QSAR) and structure-property relationships (QSPR) methods. This article focusses on the computation of analytical expressions for nine degree-based topological descriptors using the M-polynomial, for three benzenoid structures: convex benzenoid system, hexabenzocoronene (HBC), and hexa-cata-hexabenzocoronene (cHBC). Besides, the expressions for a few recently developed indices are also calculated using edge-partition techniques. These derived indices are then compared graphically based on their numerical values. This analysis of degree-based descriptors for these structures can lay the groundwork for future researchers to discover and explore new benzenoids and their properties. [ABSTRACT FROM AUTHOR]

The main focus of this paper is to develop certain types of fundamental theorems using q, q (α) , and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, q (α) , and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and q (α) are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and q (α) symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and q (α) difference equations. [ABSTRACT FROM AUTHOR]

This paper aims to compute the degree(valence)-based molecular descriptors for coronoid polycyclic aromatic hydrocarbons. The myriad types of coronoids have potential far-reaching applications in evolving fields such as nanotechnology and synthetic organic chemistry due to their superaromaticity and other extraordinary geometric and electronic properties. The degree-based connectivity descriptors, like the Randić, the sum-connectivity, the first and second Zagreb indices, and its co-indices have a strong correlation with the total π-electron energy and other properties which are significant for the synthesis of coronoid polycyclic aromatic compounds. In this article, the various degree(valence)-based topological descriptors are computed using the M-polynomial for coronoid structures with cavities that vary depending on their topology, which will benefit future researchers in exploring and synthesizing new coronoid structures. The analytical expressions for recently developed molecular descriptors are also derived. The degree-based indices have not been studied earlier for coronoid structures, which contributes substantial weightage to this research work. Additionally, the computed degree-based indices are compared both graphically and numerically. [ABSTRACT FROM AUTHOR]

Curcumin (C 21 H 20 O 6 ) is a polyphenol found in the plant Curcuma longa . Even though it possesses many pharmacological effects, owing to its limited intestinal absorption, solubility, and oral bioavailability, it is more often used as a health supplement than as a lead chemical. The poly(amido)amine (PAMAM) dendrimer (nanostructure) is utilized to enhance the stability and targeted delivery of drugs. Recently, curcumin was conjugated with the PAMAM dendrimer and analyzed for its photostability. Further investigation into the physiochemical characteristics of different generations can facilitate curcumins' targeted delivery for many diseases, including cancer. However, many of these conjugates' physiochemical properties are not available in databases since they have not been explored theoretically or experimentally. In this article, QSAR/QSPR (quantitative structure-activity relationship/quantitative structure-property relationship) analysis of physiochemical properties was carried out for component structures, which produced encouraging results. Hence, 16 discrete adriatic topological indices and their associated entropy measures were evaluated to theoretically predict a few physiochemical properties of the conjugated structure. The predictions will aid the chemist in drug designing., Competing Interests: The authors declare no competing financial interest., (© 2024 The Authors. Published by American Chemical Society.)

In the domains of materials and chemical and physical sciences, a significant aspiration is to design and synthesize extensively conjugated macrocycles possessing precisely defined structures. This objective bears substantial promise across a wide range of scientific and technological fields. These molecules offer a unique blend of structural complexity and electronic properties that make them particularly intriguing for both theoretical and practical reasons. Cycloparaphenylene (CPP) radial π-conjugated macrocycles is a specific example of a conjugated macrocycle that has garnered significant attention in the field of chemistry and materials science. It consists of a series of benzene rings linked together in a cyclic arrangement, forming a one-dimensional structure. CPP systems have been on the rise due to their novel and captivating characteristics, encompassing properties, such as electronic properties, heightened electrical conductivity, optoelectronic traits, and mechanical properties. Given the potential applications of CPP, it becomes essential to analyze this structure from a theoretical standpoint. Molecular descriptors play a crucial role in the theoretical analysis of such structures. Research on molecular descriptors has unequivocally demonstrated their significant correlation with the diverse properties of chemical compounds. This article illustrates the neighborhood sum M -polynomial-based descriptors' calculation using edge-partition techniques for CPP and its sidewalls consisting of pyrene and hexabenzocoronene units. The examination of these neighborhood sum M -polynomial-based descriptors for these structures has the potential to establish a foundational framework for delving deeper into CPP and its associated properties., Competing Interests: The authors declare no competing financial interest., (© 2023 The Authors. Published by American Chemical Society.)