Geometric-arithmetic Index Research Articles

On the ABC and GA indices of the corona products of some graphs

In this research, we derived the formulae for the atom-bond connectivity (ABC) index and the geometric-arithmetic (GA) index of several corona products of graphs made by composing the path, the cycle and the complete graphs. Relevant mathematical results were presented with proofs, indicating the possibility of predicting physicochemical properties of several molecular graphs.

Degree-Based Topological Indices of Boron B12

Topological index sometimes called molecular descriptor is a numerical value which associates a chemical composition for correlating chemical structure with numerous physical properties, chemical reactivity, or biological activity. In this paper, we study some topological indices of boron and try to correlate the physicochemical properties such as freezing points, boiling points, melting points, infrared spectrum, electronic parameters, viscosity, and density of chemical graphs. We discuss these topological indices, and some of them are mentioned here such as Randic index, the first general Zagreb index, the general sum connectivity index, hyper-Zagreb index (HM), the atom-bond connectivity index (ABC), the geometric-arithmetic index (GA), the harmonic index (H), and the forgotten index (F).

A study of newly defined degree-based topological indices via M-polynomial of Jahangir graph

There is an incredible importance of topological indices in the field of graph theory. M-polynomial is a very effective way for finding the topological indices of a graph. In this article, some important topological indices such as Atom bond connectivity index, geometric-arithmetic index, K-Banhatti indices, Hyper K-Banhatti indices and Modified K-Banhatti indices of Jahangir graph has been calculated from its M-polynomial.

On Topological Indices of Remdesivir Compound Used in Treatment of Corona Virus (COVID 19)

Corona virus cause diseases ranging from the common cold to Severe Acute Respiratory Syndrome (SARS). Several therapeutic agents have been evaluated for the treatment of Covid-19, but none have yet been shown to be efficacious. Remdesivir Gilead Sciences number (GS-5734), an inhibitor of the viral RNA-dependent, RNA polymerase with inhibitory activity against SARS-CoV and the Middle East respiratory syndrome (MERS-CoV), was identified early as a promising therapeutic candidate for Covid-19 because of its ability to inhibit SARS-CoV-2 invitro. Knowledge about the potential efficacy of Remdesivir against corona viruses has been restricted to in vitro studies and animal models. However, information related to COVID-19 is rapidly growing. Several clinical trials are ongoing for the management of COVID-19 using Remdesivir. Efficacy of Remdesivir (GS-5734) against Ebola virus has been demonstrated. In the field of Medical Science, concerning the definition of the topological index on the molecular structure and corresponding medical, biological, chemical, pharmaceutical properties of drugs can be studied by the topological index calculation. In this paper, we compute some reverse topological indices namely, the reverse general Randic index, the reverse atom bond connectivity index, the reverse geometric arithmetic index, the reverse forgotten index, the reverse Balaban index and the reverse Zagreb type indices.

Topological aspects of 2D structures of trans-[formula omitted] lattice and a metal-organic superlattice

In quantitative structure property (QSPR) and quantitative structure activity (QSAR) relationships studies, topological indices are utilized to associate the biological activity of underline structures with their physical properties like strain energy, distortion, melting point, boiling point, and stability, etc. In these studies, the degree based topological indices have obtained center position between various kind of descriptors, due to the simplicity of finding and the rapidity with which these estimations can be accomplished. In this paper, we compute the general Randic, first and second Zagreb, Atom Bond Connectivity(ABC) and Geometric Arithmetic(GA) indices of the 2D structures of trans-Pd-(NH2)S lattice and a metal–organic superlattice. Furthermore, we present the precise formulas of the fifth version of Geometric Arithmetic index and the fourth version of Atom-Bond Connectivity index of these lattices.

M-polynomial and topological indices of some transformed networks

<abstract><p>In the chemical industry, topological indices play an important role in defining the properties of chemical compounds. They are numerical parameters and structure invariant. It is a proven fact by scientists that topological properties are influential tools for interconnection networks. In this paper, we will use stellation, medial and bounded dual operations to build transformed networks from zigzag and triangular benzenoid structures. Using M-polynomial, we compute the first and second Zagreb indices, second modified Zagreb indices, symmetric division index, general Randic index, reciprocal general Randic index. We also calculate atomic bond connectivity index, geometric arithmetic index, harmonic index, first and second Gourava indices, first and second hyper Gourava indices.</p></abstract>

Some degree-based topological indices of caboxy-terminated dendritic macromolecule

Abstract In the modern era of the chemical science, the chemical graph theory has contributed significantly to exploring the properties of the chemical compounds. Currently, the computation of the topological indices is one of the most active directions of the research in the area of the chemical graph theory. The main feature of the study of the topological indices is its its ability of predicting the various physio-chemical properties. In this article, we compute several degree-based topological indices for the caboxy-terminated dendritic macromolecule. We compute Harmonic index, atom-bond connectivity index, geometric arithmetic index, sum connectivity index, inverse sum index, symmetric division degree, and Zagreb indices for caboxy-terminated dendritic macromolecule. The obtained results have potential to predict biochemical properties such as viscosity, entropy, and boiling point.

Generalization of New Degree Based Topological Indices of Silicate Network Graph

In the fields of chemical graph theory and mathematical chemistry, a topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph. Topological indices are used for example in the development of quantitative structure-activity relationships (QSARs) and other properties of molecules are correlated with their chemical structure, in that Silicates are the most common minerals in the Earth’s crust and mantle, it has a wide variety of physical properties which paved way to find the topological indices. In this paper gives us the generalized new form of topological indices such as Arithmetic-Geometric Index (AG Index), SK Index, SK1 Index and SK2 Index of Silicate network graphs.

On symmetric division deg index of trees with given parameters

Recently, the symmetric division deg ($ SDD $) index is proven to be a potentially useful molecular descriptor in QSAR and QSPR (quantitative structure-activity and structure-property relationships) studies. And its predictive capability is better than that of some popular topological indices, such as the famous geometric-arithmetic index and the second Zagreb index. In this work, the maximum $ SDD $ indices of trees with given matching number or domination number or independence number or number of pendant vertices or segments or diameter or radius are presented. Furthermore, the corresponding extremal trees are identified.

Mathematical Properties of Variable Topological Indices

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices Aα and Bα, defined for each graph H=(V(H),E(H)) by Aα(H)=∑ij∈E(H)f(di,dj)α and Bα(H)=∑i∈V(H)h(di)α, where di denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from Aα and Bα by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of Aα (respectively, Bα) involving Aβ (respectively, Bβ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

Molecular topological indices-based analysis of thermodynamic properties of graphitic carbon nitride

Graphitic carbon nitride $$({\text {g-C}}_{3}{\text {N}}_{4})$$ has become a new interdisciplinary research hot spot due to its metal-free and visible-light-responsive nature along with alluring electronic band structure and high physicochemical stability. This fascinating semiconductor conjugated polymer is considered a feasible material for harvesting clean and inexhaustible solar energy in the field of environmental remediation and energy applications. Chemical graph theory plays an important role in modeling and designing the chemical structure of complex compounds. The molecular topological indices are the numerical invariants of a molecular graph and are very useful for predicting the reactivity of respective compound. In this paper, we studied the chemical graph of crystalline graphitic carbon nitride, i.e., $$({\text {g-C}}_{3}{\text {N}}_{4})$$ , via degree-based topological indices such as general Randic index, atom bond connectivity index, geometric arithmetic index and Zagreb-type indices. Curve fitting between the calculated thermodynamic properties and topological indices was done together with the numerical and graphical representation of computed results. This mathematical framework of graphitic carbon nitride based on topological indices presents a comprehensive picture of potentially valuable thermodynamic properties such as entropy and enthalpy which could be useful in structural modification of $$({\text {g-C}}_{3}{\text {N}}_{4})$$ for specific application.

On Topological Indices of mth Chain Hex-Derived Network of Third Type

In theoretical chemistry, the numerical parameters that are used to characterize the molecular topology of graphs are called topological indices. Several physical and chemical properties like boiling point, entropy, heat formation, and vaporization enthalpy of chemical compounds can be determined through these topological indices. Graph theory has a considerable use in evaluating the relation of various topological indices of some derived graphs. In this article, we will compute the topological indices like Randić, first Zagreb, harmonic, augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices for chain hex-derived network of type 3 CHDN3(m,n) for different cases of m and n. We will also compute the numerical computation and graphical view to justify our results.Mathematics Subject Classification: 05C12, 05C90

Eccentric topological properties of a graph associated to a finite dimensional vector space

Abstract A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.

On the geometric-arithmetic Estrada index of graphs

The Estrada index and geometric-arithmetic index are two representative topological indices, and have been extensively utilized in QSPR/QSAR research. In this paper, we construct the geometric-arithmetic Estrada index EEGA, which is defined as the sum of terms eσk (1 ≤ k ≤ n), where σk are eigenvalues of the geometric-arithmetic matrix of an n-vertex graph G. First, we give some bounds for the geometric-arithmetic Estrada index, and characterize their corresponding extremal graphs. In addition, some connections between EEGA and the geometric-arithmetic energy of graphs (EGA) are determined.

Arithmetic–geometric index and its relations with geometric–arithmetic index

The arithmetic–geometric index (AG(G)) was recently introduced as a modification of the well-known geometric–arithmetic index (GA(G)). This paper reports results on searching for extremal AG-graphs for various classes of simple graphs. Additionally, relations between these two indices are elaborated. Results on combinations AG+GA,AG−GA,AG · GA, and AG/GA are given. The paper is concluded with four conjectures that have been derived based on computer investigations.

Computational Properties of General Indices on Random Networks

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M1α(G) and M2α(G), and the general sum-connectivity index, χα(G)) as well as of general versions of indices of interest: the general inverse sum indeg index ISIα(G) and the general first geometric-arithmetic index GAα(G) (with α∈R). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks GER(nER,p) and random geometric (RG) graphs GRG(nRG,r). The ER random networks are formed by nER vertices connected independently with probability p∈[0,1]; while the RG graphs consist of nRG vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius r∈[0,2]. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree k of the corresponding random network models, where kER=(nER−1)p and kRG=(nRG−1)(πr2−8r3/3+r4/2). That is, X(GER)/nER≈X(GRG)/nRG if kER=kRG, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.

Quality testing of distance-based molecular descriptors for benzenoid hydrocarbons

In this paper, we investigate the prediction power of all the well-known distance-based molecular structure descriptors in the literature for 22 lower polycyclic aromatic hydrocarbons (PAHs). The standard enthalpy/heat of formation is chosen to be the representative of thermal properties, whereas, the normal boiling point is selected to represent intermolecular and van-der-Waals type interactions. First, we present a computational technique to compute distance-based, eccentricity-based, degree-distance-based and degree based topological descriptors. The proposed method extends certain existing techniques in the literature. The proposed method is used to compute various distance-based descriptors for the 22 PAHs and then we develop the regression models with their normal boiling point and standard enthalpy of formation. Experimental results reveal some unexpected outcomes as the correlation ability of some well-known distance-based descriptors such as the Wiener and the Szeged indices is rather weak. Unlike their reputation among researchers, the second atomic-bond connectivity index and the fourth geometric-arithmetic index yield the best performance with correlation coefficients greater than 0.95. The results warrant further usage of the second atom-bond connectivity index in the structure-activity and structure-property models. Our study contributes towards slowing down the proliferation of distance-based topological descriptors.

Distance‐based topological descriptors for measuring the π‐electronic energy of benzenoid hydrocarbons with applications to carbon nanotubes

In this paper, we provide efficient regression models for correlating the π‐electronic energies of carbon nanotubes and nanocones. First, a computational technique for determining distance‐based, eccentricity‐based, degree‐distance‐based, and degree‐based molecular descriptors is proposed. Then, the application of our technique has been explained for the family of fibonacenes. Importantly, we use the proposed computational technique to determine commonly occurring distance‐based molecular descriptors and generate regression models to determine their correlation with the π‐electronic energies of lower polycyclic aromatic hydrocarbons (PAHs). Unlike its reputation among chemical graph theorists and reticular chemists, the fourth version of the geometric–arithmetic index outperforms all the distance‐based descriptors having the correlation coefficient 0.999. This warrants further usage of the fourth geometric–arithmetic index in quantitative structure‐activity relationship models. To ensure the applicability of our proposed study, we use the proposed computational technique to compute analytically explicit expressions for certain distance‐based molecular descriptors for certain infinite families of carbon nanotubes and carbon nanocones. Our results assist in correlating the π‐electronic energies of underlying chemical structures of these nanotubes and nanocones.

On the difference between geometric-arithmetic index and atom-bond connectivity index for trees

On the difference between geometric-arithmetic index and atom-bond connectivity index for trees

On Valency-Based Molecular Topological Descriptors of Subdivision Vertex-Edge Join of Three Graphs

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In these studies, degree-based topological indices have a significant place among the other descriptors because of the ease of generation and the speed with which these computations can be accomplished. In this paper, we give the results related to the first, second, and third Zagreb indices, forgotten index, hyper Zagreb index, reduced first and second Zagreb indices, multiplicative Zagreb indices, redefined version of Zagreb indices, first reformulated Zagreb index, harmonic index, atom-bond connectivity index, geometric-arithmetic index, and reduced reciprocal Randić index of a new graph operation named as “subdivision vertex-edge join” of three graphs.