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In this study, twelve drugs used to treat Parkinson's disease were analyzed. To simplify calculations and data analysis, a computer-based computing technique along with the algorithm has been employed. The M-polynomial and their degree-based topological indices derived from the M-polynomial were calculated using MATLAB coding. Linear, quadratic, cubic, logarithmic, inverse, power, compound, s-curve, growth, and exponential regression model analyses were utilized to create QSPR models between the topological indices and eight the Physicochemical properties of the drugs to determine their effectiveness. Confidence intervals at a 95% level were computed for both the slope and intercept of the linear regression models. Also, based on the maximum $ R^{2} $ R 2 , optimal equations for estimating the Boiling point, Enthalpy of vaporization, Molar refractivity, Polarizability, and Molar volume using different indices have been determined, and linear, quadratic, and cubic equations have been specified. Calculated feature values are strongly correlated with actual values, indicating reliable predictive capabilities of the indices. For statistical analysis and to determine if there is a significant difference between the averages of the two groups, we used either an independent T-test or Welch's T-test. The results show that the p-value is less than 0.05, which indicates that the mean difference between the samples is statistically significant. [ABSTRACT FROM AUTHOR]

Topological indices are often used to predict the physicochemical properties of molecules. The multiplicative sum Zagreb index is one of the multiplicative versions of the Zagreb indices, which belong to the class of most-examined topological indices. For a graph G with edge set E = { e 1 , e 2 , ⋯ , e m } , its multiplicative sum Zagreb index is defined as the product of the numbers D (e 1) , D (e 2) , ⋯ , D (e m) , where D (e i) is the sum of the degrees of the end vertices of e i . A chemical tree is a tree of maximum degree at most 4. In this research work, graphs possessing the maximum multiplicative sum Zagreb index are determined from the class of chemical trees with a given order and fixed number of segments. The values of the multiplicative sum Zagreb index of the obtained extremal trees are also obtained. [ABSTRACT FROM AUTHOR]

A QSPR (Quantitative Structure-Property Relationship) model links mathematically various physicochemical properties with the structure of a molecule. Establishing such quantitative relationships is of great technological importance as in this way one can predict the properties of new untested molecules by means of a linear or nonlinear equation that expresses a certain property as an explicit function of one or more independent variables. These functional relationships (usually called QSPR models) are obtained by performing specific QSPR studies on a class of similar compounds whose properties have been already determined and were appropriately correlated to their molecular structures. In this study we are interested to determine the best QSPR models for the normal boiling point (bp) and molar refraction (mr) of 64 alkyl halides by using the topological index ZEP, the structural parameter (Hd) and the number of carbon atoms (N) as predictor variables. In order to validate the developed models with respect to the goodness-of-fit, robustness and predictive ability for the two properties considered in the class of alkyl halides, external validation, cross-validation (leave-one-out) and randomization (y-randomization) were performed. The obtained results have shown that the three descriptors above could be efficiently used for modelling and predicting the normal boiling points and molar refractions of the considered class of compounds. [ABSTRACT FROM AUTHOR]

Let G be a graph and S be a subset of V (G). A graph G is called S -leaf-connected if G has a spanning tree T such that S is the set of leaves of T. For k ≥ 2 , we say a graph G is k -leaf-connected if | V (G) | > k and given any subset S of V (G) with | S | = k , G is S -leaf-connected. In the present note, we give some sufficient conditions on eccentric connectivity index, eccentric distance sum, connective eccentricity index and difference of Zagreb indices for connected graphs to be k -leaf-connected. These results generalize the sufficient conditions on these indices for connected graphs to be Hamilton-connected. At last, we obtain a sufficient condition for a graph to be k -leaf-connected in terms of the general sum-connectivity index of the complement graph. [ABSTRACT FROM AUTHOR]

A nanotube is a nanometer–scale tube-like structure, it is a kind of nanoparticle, and may be large enough to serve as a pipe through which other nanoparticles can be channeled, or, depending on the material, may be used as an electrical conductor or an electrical insulator. For computing the structural information of nanotubes, the graph entropies have become the information theoretic quantities. The graph entropy measure has attracted the research community due to its potential application in discrete mathematics, biology, and chemistry. In this paper, our contribution is to explore graph entropies for structures of some nanotubes based on novel information function, which is the number of different degree vertices along with the number of edges between various degree vertices. More precisely, we computed entropies of some classes of nanotubes such as titania nanotube TNT 3 [ m , n ] , TNT 6 [ m , n ] and carbon nanotubes HAC 5 C 6 C 7 [ m , n ] by making a relation of degree-based topological indices with the help of information function. [ABSTRACT FROM AUTHOR]

In Sri Lanka as well as the rest of the globe, cancer is the top cause of mortality. One of the key medicines in treating tumors is anticancer medications and delivery dendrimers. To prevent the formation of the rapid proliferation of cancer cells, several tests were carried out. Because of this, research on dendrimers and anti-cancer medications is crucial. Topological indices (TIs) are molecular descriptors numerical values corresponding to the physical characteristics of a molecule's chemical structure. It costs money to determine a molecule's physical characteristics in a lab since it takes a lot of materials, medications, and time. Therefore, the relevant information about molecules may be obtained by computing TIs. This study's goals are to compute hitherto uncalculated eccentricity-based TIs for various anticancer structures and to use curvilinear regression models to forecast the physical characteristics of particular anticancer medications. These anticancer medications were given different TIs developed in this work, allowing the researchers to understand the physical, physicochemical, and chemical characteristics related to them. In addition comparative study of the novel indices with some well-known and mostly used indices in structure–property modeling and anticancer drugs in performed. [ABSTRACT FROM AUTHOR]

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds. [ABSTRACT FROM AUTHOR]

A novel geometric method is proposed for constructing vertex‐degree‐based molecular structure descriptors (topological indices). The model is based on an ellipse whose focal points represent the degrees of a pair of adjacent vertices. This approach enables a geometric interpretation of several previously known topological indices, and lead to design of a few new. The area of the ellipse induces a vertex‐degree‐based topological index of remarkable simplicity, which we call elliptic Sombor index(ESO$$ ESO $$). The main mathematical properties of ESO$$ ESO $$ are established, especially its relations to other, earlier known, indices. Then the applicative potential of ESO$$ ESO $$ is analyzed. [ABSTRACT FROM AUTHOR]

Topological indices are numerical parameters derived from the structural information of chemical compounds. By providing a quantitative description of molecular structures, topological indices enable researchers to predict various properties and behaviors of molecules. The Randić index (R$$ R $$) and the geometric–arithmetic index (GA$$ GA $$) are widely recognized topological indices. It is observed that, GA(G)>R(G)$$ GA(G)>R(G) $$ for any given graph G$$ G $$. We aim to investigate the gap between GA$$ GA $$ and R$$ R $$ for chemical tree. The complete characterization of minimal chemical tree for GA−R$$ GA-R $$ is carried out here. This article offers an interesting finding that, whereas GA$$ GA $$ and R$$ R $$ provide same minimal chemical trees, GA−R$$ GA-R $$ yields minimal tree structures that are totally different from GA$$ GA $$ and R$$ R $$. Moreover GA−R$$ GA-R $$ is observed to correlate well with physico‐chemical properties of octanes. [ABSTRACT FROM AUTHOR]

This study delves into a comprehensive physical analysis of entropy measures applied to sodium oxide Na2O. A key idea in information theory and thermodynamics, entropy is essential to comprehending the stability of a system. The study clarifies the intricate relationships and physical properties of Na2O by combining theoretical analysis with statistical methods, providing important knowledge for material design and industrial operations. In a chemical graph, atoms are shown by vertices while their bonding are illustrated by edges. Sodium oxide is a major contributor to manufacture glass, it also has potential applications in CO2 sequestration, transparent materials, biomedical devices and nano grating glass. A topological index is a relationship between the molecular graph and its topology. We have computed various graph entropies based on different topological indices of chemical graph of sodium oxide. Further we have integrated these graph entropies with distinct thermodynamical measures of sodium oxide by developing mathematical models between both quantities. We have developed these mathematical frameworks in MATLAB. All the models are selected relying on the least mean squared error or sum of squared error. [ABSTRACT FROM AUTHOR]

In this study, various beta‐blocker drugs used for heart disease were analyzed, and their degree‐based topological indices derived from the M‐polynomial were calculated. Linear and quadratic regression analysis was used to obtain quantitative structure‐property relationship models between the topological indices and eight the physicochemical properties of the drugs to determine their effectiveness. The results show that the harmonic index was the best predictor for boiling point, flashpoint, and enthalpy of vaporization, while the redefined third Zagreb index was effective for polarizability, molar refractivity, and molar volume. The inverse sum indeg index was found to be effective for molar refractivity, and the second modified Zagreb index was surface tension in linear regression models. In addition, the redefined third Zagreb index was the best predictor for polarizability and molar refractivity, while the second modified Zagreb index was effective for molar volume. The SDD index was found to be effective for surface tension in quadratic regression models. [ABSTRACT FROM AUTHOR]

The structural features of polycyclic aromatic compounds, like the fundamental molecular topology, strongly influence their chemical and biological properties. Quantitative structure-activity and property relationship (QSAR/QSPR) techniques for predicting characteristics of polycyclic aromatic compounds (PAC) and related graphs from chemical structures have been developed in this approach. To obtain degree-based topological indices, we have many polynomials. The neighborhood M-polynomial is one of these polynomials used to produce several topological indices based on neighborhood degree sum. In this study, we offer the exact analytical expressions of neighborhood M-polynomial and their corresponding topological indices for supercoronene (SC), cove-hexabenzocoronene (cHBC), and triangular-shaped discotic graphene (TDG) with hexabenzocorenene (HBC) as the base molecule. The findings could help with the development of characteristic physicochemical prediction. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of Z n , where n = p α , p 2 q , p 2 q 2 , p q r , p 3 q , p 2 q r , and p q r s for the different prime integers p , q , r , and s. Moreover, we construct M-polynomials and C o M -polynomials using the P I S -graph structure of Z n to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and C o M -polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory. [ABSTRACT FROM AUTHOR]

Abstract Approaches based on graph descriptors have been used in cheminformatics and bioinformatics for molecular property prediction. In this article, we apply partition technique and simple counting schemes to describe the structure of a two-dimensional (2D) kagome graphene and then derive the closed form of various degree-based topological indices (graph descriptor) of 2D kagome graphene, and its tubular and toroidal forms. [ABSTRACT FROM AUTHOR]

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set { x 1 , ... , x n } , then the ad-hoc Lanzhou index of G is defined by L z ˜ (G) = ∑ i = 1 n d i (n − 1 − d i) 2 , where d i represents the degree of the vertex x i . Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ -cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ -cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order. [ABSTRACT FROM AUTHOR]

A QSPR analysis is carried for square root stress-sum index of molecular graphs and physical properties of lower alkanes and linear regression models are presented for boiling points, molar volumes, molar refractions, heats of vaporization and critical temperatures. [ABSTRACT FROM AUTHOR]

The chemical graph theory helps in understanding the complex structure of molecules. Researchers can acquire a detailed grasp of the physical science, chemical properties, and bio-organic aspects of pharmaceuticals by computing the resolvability, and topological parameters of a drug design. The resolvability constraints for graph G = (V , E) are a relatively new advanced field in which the entire structure is constructed so that each vertex (atom) or edge (bond) indicates a distinct position. A topological index is a type of molecular descriptor that is effectively used in modelling many physicochemical properties in numerous quantitative structure–property/activity relationship (QSPR/QSAR) studies. In the QSPR investigation, topological indices have been used to forecast the bioactivity of chemical molecules. In the proposed study, the metric, edge metric resolvability, and topological indices of oseltamivir and zanamivir, two extensively used antiviral medicines, are computed. Further, we also discuss linear regression analysis between indices and physiochemical properties of drugs. This theoretical analysis may help the chemist and people working in the pharmaceutical industry to predict the properties of zanamivir and oseltamivir drugs without experimenting. [ABSTRACT FROM AUTHOR]

We study a boundary value problem for a Caputo‐type fractional differential equation subjected to periodic boundary conditions. For an auxiliary problem with the simplified right‐hand side, we explicitly construct its unique solution. In addition, based on the theory of the topological index, we prove existence of at least one solution to the original problem. [ABSTRACT FROM AUTHOR]

Totally made of carbon atoms, fullerene is a spherical molecule with a three-dimensional pattern like a soccer ball. The most prevalent kind of fullerene is C60, which has 60 carbon atoms organised in a pattern like that of a soccer ball (12 pentagons and 20 hexagons). The carbon atoms are connected by a web of covalent connections to form what is known as a truncated icosahedron. The fullerene molecule's spherical form makes it beneficial in many applications, including medication delivery, lubricants, and even as a possible component of nanotechnology. Topological indices or numerical descriptors have been used for decades to connect essential physicochemical parameters with significant molecular structural features including cyclicity, boiling and melting point, and enthalpy for a range of two- along with three-dimensional in form chemical arrangement networking or graphs. The valency of atom-based numerical descriptors in this study are computed using the valency of atom in a chemical network or structure of the molecular investigation. In chemoinformatics, the numerical descriptors based on the degree of defined by reverse-degree concept technique is a more newly developed methodology for evaluating chemical systems and geometries. It draws attention to molecular traits as numerical descriptors based on the degree of defined by reverse-degree concept and presents numerical descriptors in algebraic form. In this work, we describe several fullerene cages as numerical descriptors based on the degree of defined by reverse-degree concept and compute a number of numerical descriptors based on the degree of defined by reverse-degree concept, such as the Randić index, atom-bond-connectivity, geometric arithmetic, forgotten, hyper, first and second Zagreb indices are computed. [ABSTRACT FROM AUTHOR]

The intriguing properties of polycyclic structure known as benzenoid hydrocarbons, which are unsaturated and totally conjugated molecules with hexagonal rings, continue to entice researchers to investigate their chemical and physical molecular structure with reference to aromaticity. To correlate important molecular structural attributes such as enthalpy, melting, boiling point, and cyclicity, numerical descriptors or topological indices which have been used for decades, for variety of two and three dimensional chemical structures networks or graphs with vital physicochemical properties. To determine the degree‐based topological indices, in this study, the degrees of the chemical structure or graph of the molecule under examination are utilized. In molecular graph theory, M‐polynomial is a relatively recent approach for investigating chemical networks and structures. It shows numerical descriptors in algebraic form and highlights molecular properties as a polynomial function. In this study, we propose a polynomials presentation of two‐dimensional coronene fractal structures and compute numerous M‐polynomials, including the M‐polynomials of the first and second Zagreb indexes, as well as the Randic index. [ABSTRACT FROM AUTHOR]

Chemical graph theory is a branch of mathematical chemistry that significantly affects the development of the chemical sciences. The neighborhood M-polynomial approach is an extraordinary process for determining neighborhood degree sum-based topological indices to estimate materials' various physical, chemical, and biological characteristics. The present work deals with obtaining the NM-polynomial of three types of polycyclic aromatic hydrocarbons (PAHs). Using NM-polynomials as tools, specific degree-based topological indices are retrieved. The applications of these PAHs are pharmaceuticals, lubricating materials, photographic products, agricultural products and other chemical industries. The graphical representations of the outcomes are displayed. These results are compared to several well-known degree-based indices. [ABSTRACT FROM AUTHOR]

Graph entropy plays an essential role in interpreting the structural information and complexity measure of a network. Let G be a graph of order n. Suppose d G (v i) is degree of the vertex v i for each i = 1 , 2 , ... , n . Now, the k-th degree-based graph entropy for G is defined as I d , k (G) = − ∑ i = 1 n d G (v i) k ∑ j = 1 n d G (v j) k l o g d G (v i) k ∑ j = 1 n d G (v j) k , where k is real number. The first-degree-based entropy is generated for k = 1 , which has been well nurtured in last few years. As ∑ j = 1 n d G (v j) k yields the well-known graph invariant first Zagreb index, the I d , k for k = 2 is worthy of investigation. We call this graph entropy as the second-degree-based entropy. The present work aims to investigate the role of I d , 2 in structure property modeling of molecules. [ABSTRACT FROM AUTHOR]

Discovering a vaccine with reliable and effective treatment for the new corona virus disease 2019 (COVID-19) is indeed a long way off, and there seems to be a critical need to research additional viable medications that could save countless lives in the event of the COVID-19 pandemic. Scientists from all over the world are actively developing medical or anti-virus medications, which are safe and effective for COVID-19. The anti-viral drugs of chloroquine (CQ) and hydroxychloroquine (HCQ) have proved to be an efficient inhibitory effect by preventing the binding of spike covid protein. The determination of a pharmaceutical structure using a topological index allows researchers to have a better understanding of the physiological as well as bio-organic properties of drugs. The goal of this study is to employ molecular graph theory to determine some graph-theoretic parameters related to the molecular graph of CQ and HCQ. In this paper, we present some resolvability parameters such as metric dimension (MD), edge metric dimension (EMD), fault-tolerant metric dimension (FTMD), and fault-tolerant edge metric dimension (FTEMD) of CQ and HCQ. We prove that these resolvability parameters for CQ and HCQ are bounded and constant. [ABSTRACT FROM AUTHOR]

This paper is concerned with the general atom-bond sum-connectivity index A B S γ , which is a generalization of the recently proposed atom-bond sum-connectivity index, where γ is any real number. For a connected graph G with more than two vertices, the number A B S γ (G) is defined as the sum of (1 − 2 (d x + d y) − 1) γ over all edges x y of the graph G, where d x and d y represent the degrees of the vertices x and y of G, respectively. For − 10 ≤ γ ≤ 10 , the significance of A B S γ is examined on the data set of twenty-five benzenoid hydrocarbons for predicting their enthalpy of formation. It is found that the predictive ability of the index A B S γ for the selected property of the considered hydrocarbons is comparable to other existing general indices of this type. The effect of the addition of an edge between two non-adjacent vertices of a graph under A B S γ is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved. [ABSTRACT FROM AUTHOR]

The quantitative characterization of the topological structures of irregular graphs has been demonstrated through several irregularity measures. In the literature, not only different chemical and physical properties can be well comprehended but also quantitative structure-activity relationship (QSPR) and quantitative structure-property relationship (QSAR) are documented through these measures. A simple graph G = (V, E) is a collection of V and E as vertex and edge sets respectively, with no multiple edges or loops. Keeping in view the importance of various irregularity measures, in (Abdo et al., 2014a) the authors defined the total irregularity of a simple graph G = G(V, E) as irrt(G) = 1 2 P u,v∈V |dG(u) - dG(v)|, where dG(u) indicates the degree of the vertex u, where u ∈ V (G). In this paper, we have determined the first minimum, second minimum and third minimum total irregularity index of the tricyclic graphs on the n vertices. [ABSTRACT FROM AUTHOR]

Several topological indices are known to have widespread implications in a variety of research areas. Over the years, the Kirchhoff index has turned out to be an extremely significant and efficient index. In this paper, we propose the exact formulas for the expected values of the random polyomino chain to construct the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index. We also carefully examine the highest degree of the expected values for a random polyomino chain through the multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index. [ABSTRACT FROM AUTHOR]

A topological index is a numeric quantity associated with a chemical structure that attempts to link the chemical structure to various physicochemical properties, chemical reactivity, or biological activity. Let R be a commutative ring with identity, and Z * (R) is the set of all non-zero zero divisors of R. Then, Γ (R) is said to be a zero-divisor graph if and only if a · b = 0 , where a , b ∈ V (Γ (R)) = Z * (R) and (a , b) ∈ E (Γ (R)) . We define a ∼ b if a · b = 0 or a = b . Then, ∼ is always reflexive and symmetric, but ∼ is usually not transitive. Then, Γ (R) is a symmetric structure measured by the ∼ in commutative rings. Here, we will draw the zero-divisor graph from commutative rings and discuss topological indices for a zero-divisor graph by vertex eccentricity. In this paper, we will compute the total eccentricity index, eccentric connectivity index, connective eccentric index, eccentricity based on the first and second Zagreb indices, Ediz eccentric connectivity index, and augmented eccentric connectivity index for the zero-divisor graph associated with commutative rings. These will help us understand the characteristics of various symmetric physical structures of finite commutative rings. [ABSTRACT FROM AUTHOR]

Let G be a graph with edge set E (G). Denote by d w the degree of a vertex w of G. The sigma index of G is defined as ∑ u v ∈ E (G) (d u − d v) 2 . A connected graph of order n and size n + k − 1 is known as a connected k -cyclic graph. Abdo et al. (2018) characterized the graphs having the greatest sigma index over the family of all connected graphs of a fixed order. The primary goal of the present note is to determine the unique graph possessing the greatest sigma index from the class of all connected k -cyclic graphs of order n for every pair of fixed integers k and n satisfying 0 ≤ k ≤ n − 2. [ABSTRACT FROM AUTHOR]

Let G = (V (G) , E (G)) be a simple molecular graph without directed and multiple edges and without loops. Graph theory has become an important component of the chemical mathematics, and it has become one of the most powerful mathematical tools in the analysis and study of the architecture of a chemical network. It studies of descriptors in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies in the chemistry science. There are a lot of topological indices in QSPR/QSAR studies. In this paper, some eccentricity-based topological indices namely the eccentric connectivity index ξ c (G) , the modified eccentric connectivity index ξ c (G) , the total eccentricity index ξ (G) , the second Zagreb eccentricity index M 1 * * (G) and the average eccentricity index avec (G) are computed for the para-line graphs of some hexagonal cactus chains namely the para-chain L n , the ortho-chain O n and the meta-chain M n. [ABSTRACT FROM AUTHOR]

The Zagreb index (ZI) of a crisp graph and also for a fuzzy graph (FG) is a very much useful tool in network theory, spectral graph theory, molecular chemistry and several fields of chemistry and mathematics. The second ZI is studied for FGs here. Bounds of this index are calculated for several FGs: path, star, cycle, complete FG, partial fuzzy subgraph, etc. For isomorphic FGs, it is shown that the value of this index is same. Bounds of this index for the Cartesian product, composition, join and union of two FGs are established. At the end of this article, an application of the index in mathematical chemistry is studied. For this, octane isomers are considered and analyzed the correlation between this index with some physico-chemical properties of octane isomers. This index’s correlation coefficient (r) with acentric factor and entropy is determined for the linear curve fittings. Using the value of r, one can conclude that this index can help estimate the acentric factor and entropy with significant accuracy. Also, these outcomes declare the appropriateness of the index in QSPR research. [ABSTRACT FROM AUTHOR]

In this brief paper, the stress-sum index of molecular graphs and the physical characteristics of lower alkanes are analysed using QSPR, and linear regression models for boiling points, molar volumes, molar refractions, heats of vaporisation and critical temperatures are presented. [ABSTRACT FROM AUTHOR]

The new Euler-Jacobi formula for double points provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar quadratic–quartic polynomial differential systems when these systems have seven finite singular points being one of them with multiplicity two. The case with eight finite singular points has already been solved using the classical Euler-Jacobi formula. [ABSTRACT FROM AUTHOR]

The branching index (also known as the connectivity index), introduced in Milan Randić's seminal paper (J Am Chem Soc 97(23):6609–6615, 1975), is one of the the most famous, investigated, and applied among the graph-theoretical molecular descriptors. The atom-bond connectivity (ABC) index (Estrada et al in Indian J Chem A 37:849–855, 1998) and the sum-connectivity (SC) index (Zhou and Trinajstić in J Math Chem 46:1252–1270, 2009) belong to the class of successful variants of the connectivity index. In the present paper, by amalgamating the core idea of the SC and ABC indices, a new molecular descriptor is put forward—the atom-bond sum-connectivity (ABS) index. The graphs attaining the extreme values of the ABS index are determined over the classes of (molecular) trees and general graphs of a fixed order. A noteworthy property of the ABC index is that it increases when a non-isolated edge is inserted between any two non-adjacent vertices. It is proved that this property holds also for the ABS index. [ABSTRACT FROM AUTHOR]

Chemical descriptors are numeric numbers that capture the whole graph structure and comprise a basic chemical structure. As a topological descriptor, it correlates with certain physical aspects in addition to its chemical representation of underlying chemical substances. In the modelling and design of any chemical network, the graph is important. A number of chemical indices have been developed in theoretical chemistry, including the Wiener index, the Randić index, and many others. In this paper, we look at the benzenoid networks and calculate the exact topological indices based on the degrees of the end vertices. [ABSTRACT FROM AUTHOR]

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph Γ G of a finite group G is a graph where non-central elements of G are its vertex set, while two different elements are edge connected when they do not commute in G. In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of SL (2 , C) . We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups. [ABSTRACT FROM AUTHOR]

In the theory of chemical graphs, several topological indices were defined or expanded or generalized by Wiener, Gutman and mathematicians and other chemists [1]. In this paper, we will calculate the indices of Szeged and vertex-PI for 4k-cycle polyomino chains. [ABSTRACT FROM AUTHOR]

The new Euler–Jacobi formula for points with multiplicity two provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar cubic polynomial differential systems when these systems have eight finite singular points, being one of them with multiplicity two. The case with nine finite singular points has already been solved using the classical Euler–Jacobi formula. [ABSTRACT FROM AUTHOR]

The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. The second law of thermodynamics establishes the concept of entropy as a physical property of a thermodynamic system. Graph entropies defined for various graph invariants have shown applications in different fields of physics, chemistry, statistics, biology, and social sciences. In this paper, we define graph entropies in terms of fifth geometric-arithmetic index, fourth atom-bond connectivity index and Sanskruti index. We computed the bounds of these graph entropies for arbitrary graphs and for molecular graphs of single-walled armchair carbon nanotube Y-junctions and constructed the regression equations between the computed entropies and topological invariants. [ABSTRACT FROM AUTHOR]

There are many different types of topological indices that have been known to exist with vast literature based on them. First of its kind (topological index) was introduced in 1947 by Harry Wiener and it also occurred in an approximate formula for the total π -electron energy of the molecule and the second Zagreb index appeared within the study of molecular branching. Recently, the first reverse Zagreb alpha, beta and second reverse Zagreb indices of a simple connected graph G are defined. The main object of this work is to study the reverse first and second Zagreb indices of the F-sum of two graphs G and H, denoted by G + FH in terms of the other indices of the original graphs G and H. [ABSTRACT FROM AUTHOR]

Given a graph G = (V , E) with vertex set V and edge set E , we extend the concept of k -matching number and Hosoya index to a weighted graph (G ; ω) , where ω is a weight function defined over E. In particular, if φ is a vertex-degree-based (VDB) topological index defined via φ = φ (G) = ∑ u v ∈ E φ d G (u) , d G (v) , where d G (u) is the degree of the vertex u and φ i , j is an appropriate function with the property φ i , j = φ j , i , then we consider the weighted graph (G ; φ) with weight function φ : E → R defined as φ (u v) = φ d G (u) , d G (v) , for all u v ∈ E. It turns out that m ((G ; φ) , 1) , the number of weighted 1-matchings in (G ; φ) , is precisely φ (G) , and for k ≥ 2 , the k -matching numbers m ((G ; φ) , k) can be viewed as new k th order VDB-Hosoya indices. Later, we consider the extremal value problem of the Hosoya index over the set T n ; φ = (T ; φ) : T ∈ T n , where T n is the set of trees with n vertices. [ABSTRACT FROM AUTHOR]

Topological indices (molecular descriptors) are numerical values of a chemical structure and represented by a graph. Molecular descriptors are used in QSPR/QSAR modeling to determine a chemical structure's physical, biological, and chemical properties. The cycle graphs are symmetric graphs for any number vertices. In this paper, recently defined neighborhood degree sum-based molecular descriptors and polynomials are studied. NM-polynomials and molecular descriptors of some cycle-related graphs, which consist of the wheel graph, gear graph, helm graph, flower graph, and friendship graph, are computed and compared. [ABSTRACT FROM AUTHOR]

The Hosoya index and the Merrifield‐Simmons index are two important molecular descriptors in chemical graph theory. The Hosoya index is defined as the total number of matchings of the graph and the Merrifield‐Simmons index is defined as the total number of independent sets of the graph. In this paper, the author obtains the extremal values of the Hosoya index and the Merrifield‐Simmons index of trees with given domination number. [ABSTRACT FROM AUTHOR]

Molecular topological indices are numerical descriptor of molecular structure obtained via molecular graph G. The forgotten topological index F(G) is a vertex degree based topological index and it can be expressed with F (G) = ∑ d (v) 3 = ∑ d (u) 2 + d (v) 2 , where d(u) denotes the degree of u. The forgotten co-index Co-F(G) is defined as the sum of squares of a graph's vertex degrees which is not adjacent. In this study, the F(G) and Co-F(G) index of HAC5C7, HAC5C6C7, linear [n]-phenylenes and cyclic phenylenes nanostructures are computed for possible works in the properties of molecules such as structure–property relationship. In addition, polynomials of the forgotten index and co-index are found for these nanostructures. [ABSTRACT FROM AUTHOR]

Let G=(E(G),V(G)) be a (molecular) graph with vertex set V(G) and edge set E(G). The forgotten Zagreb index and the hyper Zagreb index of G are defined by F(G)=∑u∈V(G)d(u)3 and HM(G)=∑uv∈E(G)(d(u)+d(v))2 where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. A recent problem called the inverse problem deals with the numerical realizations of topological indices. We see that there exist trees for all even positive integers with F(G)>88 and with HM(G)>158. Along with the result, we show that there exist no trees with F(G)<90 and HM(G)<160 with some exceptional even positive integers and hence characterize the forgotten Zagreb index and the hyper Zagreb index for trees. [ABSTRACT FROM AUTHOR]

The novel coronavirus disease 2019 (COVID-19) emerged in Wuhan, China, and has spread rapidly to nearly every part of the world. Unfortunately, no drug or vaccine has been accepted for the treatment of this pandemic. Researchers have established the efficacy of some existing antiviral drugs to control COVID-19 in vitro. Some of them are remdesivir (GS-5734), chloroquine, hydroxychloroquine, theaflavin. Topological indices are mathematical interpretations of a molecule generated by an algorithm implemented to a given molecular representation. Topological indices are used to model different physicochemical properties and biological activities of chemical compounds. In this work, some degree-based and neighborhood degree sum-based topological indices are investigated for the aforesaid antiviral drugs using polynomial approach. The results obtained can aid in the design of new medicine for the treatment of COVID-19. [ABSTRACT FROM AUTHOR]

A graph is a tool for designing a system's required interconnection network. The topology of such networks determines their compatibility. For the first time, in this work we construct subdivided η ζ network S (η ζ Γ) and discussed their topology. In graph theory, there are a variety of invariants to study the topology of a network, but topological indices are designed in such a way that these may transform the graph into a numeric value. In this work, we study S (η ζ Γ) via Zagreb connection indices. Due to their predictive potential for enthalpy, entropy, and acentric factor, these indices gain value in the field of chemical graph theory in a very short time. η ζ Γ formed by ζ time repeated process which consists η ζ copies of graph Γ along with η 2 | V (Γ) | ζ η ζ − 1 edges which used to join these η ζ copies of Γ. The free hand to choose the initial graph Γ for desired network S (η ζ Γ) and its relation with chemical networks along with the repute of Zagreb connection indices enhance the worth of this study. These computations are theoretically innovative and aid topological characterization of S (η ζ Γ) . [ABSTRACT FROM AUTHOR]

Recent findings of diverse forms of carbon nanostructures have encouraged research on their applications in several areas. They are used as a structural material in electronic and optical devices, involving transistors, photodetectors, plastic, biological sensors, pharmaceutical and medicine, and drug delivery systems. Since the discovery of Y‐shaped carbon nanotubes (CNTs) junctions, they have drawn interest for future electron devices, for example, three‐terminal transistors, amplifiers, and switches. A topological index is a number that is used to study the physicochemical properties of chemical structures. In this paper, we are providing topological properties of CNT Y‐junctions. The results obtained for different types of Y‐junction graphs appeared with comparisons. According to the best of our knowledge and literature, this is the first report about topological indices on this class of CNT junctions. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with vertex set V and edge set E , and let d i be the degree of the vertex v i ∈ V. If the vertices v i and v j are adjacent, we denote the respective edge by v i v j ∈ E. A vertex-degree-based topological index φ is defined as φ (G) = ∑ v i v j ∈ E φ d i , d j , where φ i , j is a function with the property φ i , j = φ j , i. The general extended adjacency matrix A φ is defined as [ A φ ] i j = φ d i , d j if v i v j ∈ E , and 0 otherwise. The energy associated to φ of G is the sum of the absolute values of the eigenvalues of A φ. In this paper we show that ρ (G) E φ (G) ≥ 2 φ (G) for all connected graphs G , where ρ (G) is the spectral radius of G and φ a , b ≠ 0 for all a , b ∈ N. We also characterize the graphs where equality holds. As a consequence, for any tree T with n vertices, E φ (T) ≥ 2 n − 1 φ 1 , (n − 1) , with equality holding if and only if T ≅ S n. [ABSTRACT FROM AUTHOR]

In this paper we study polynomial vector fields on C 2 of degree larger than 2 with n 2 isolated singularities. More precisely, we show that if two polynomial vector fields share n 2 - 1 singularities with the same spectra (trace and determinant) and from these singularities n 2 - 2 have the same positions, then both vector fields have identical position and spectra at all the singularities. Moreover we also show that if two polynomial vector fields share n 2 - 1 singularities with the same positions and from these singularities n 2 - 2 have the same spectra, then both vector fields have identical position and spectra at all the singularities. Moreover we also prove that generic vector fields of degree n > 2 have no twins and that for any n > 2 there exist two uniparametric families of twin vector fields, i.e. two different families of vector fields having exactly the same singular points and for each singular point both vector fields have the same spectrum. [ABSTRACT FROM AUTHOR]

This research article aims to establish the values of fourteen notable topological indices that are defined on some chemical structures belonging to the Penicillin class of drugs, which are a type of antibiotic derived from Penicillin fungi. This is performed to predict the Physicochemical properties of these compounds through Quantitative Structure-Property Relationships (QSPR). Also, it is observed that these characteristics have a good correlation with the Physico-chemical characteristics of the selected chemical structures of the penicillin class of compounds. The chemical properties of the selected compounds that this paper discusses include Polarizability, Log Octanol-Water Partition Coefficient(Log Kow), Molecular Volume, Molar Refractivity, Log P, Soil Absorption Coefficient (Koc), Log Koc. [ABSTRACT FROM AUTHOR]