Geometric-arithmetic Index Research Articles

GENERALIZED REDEFINED ZAGREB INDEX ON SOME NANOSTRUCTURES

Chemical graph theory has become a significant part of the research. Topological indices are numerical descriptors used to determine physico-chemical properties of chemical compounds. In this work, the generalized redefined Zagreb index is defined. This index is applied on several topological indices viz., redefined Zagreb indices (ReZG1(G), ReZG2(G), ReZG3(G)), sum-connectivity Zagreb index (SCI(G)), hyper Zagreb index (HM(G)), harmonic index (H(G)), arithmetic-geometric index (AGI(G)), SK index (SK(G)) and SK1 index (SK1(G)). The above said indices are calculated for aromatic compounds such as boron triangular sheet, borophene chain and melem chain nanostructures. The obtained results are useful in understanding the behaviour of compounds under the study.

Topological Evaluation of Four Para-Line Graphs Absolute Pentacene Graphs Using Topological Indices

A real-number to molecular structure mapping is a topological index. It is a graph invariant method for describing physico-chemical properties of molecular structures specific substances. In that article, We examined pentacene’s chemical composition. The research on the subsequent indices is reflected in our paper, we conducted an analysis of several indices including general randic connectivity index, first general zagreb index, general sum-connectivity index, atomic bond connectivity index, geometric-arithmetic index, fifth class of geometric-arithmetic indices, hyper-zagreb index, first and second multiple zagreb indices for a four para-lines graphs of linear [n]-pentacene and multi-pentacene.

Study of eccentricity based topological indices for benzenoid structure

Topological indices play paramount role in defining chemical and structural properties of different compounds. Chemical graph theory is useful for predicting the bioactivity and physicochemical properties through numerical invariants. A broad range of topological indices are studied and used in theoretical chemistry. The eccentricity of a graph is maximum distance from one vertex a to another vertex b. The connectivity based topological indices depict vital role in mathematical chemistry. In this paper, we discuss Benzenoid Hourglass Network and calculate its total eccentricity, average eccentricity, atom-bond connectivity index, eccentricity based Zagreb index and geometric-arithmetic index. Some new helpful closed formulas are developed.

Reverse-degree-based topological indices of fullerene cage networks

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.

Semi-Total Point Graph of Neighbourhood Edge Corona Graph of G and H

A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. Here we concentrated on topological indices involving the number of vertices, the number of edges and the maximum and minimum vertex degree. The aim of this paper is to compute the lower and upper bounds of the second Zagreb index, third Zagreb index, Hyper Zagreb index, Harmonic index, Redefined first Zagreb index, First reformulated Zagreb index, Forgotten topological index, square F-index, Sum-connectivity index, Randic index, Reciprocal Randic index, Gourava index, Sombar index, Nirmala index, Geometric-Arithmetic index and lower bonds of Atom bond connectivity index, Redefined second Zagreb.

Bounds for the geometric–arithmetic index of unicyclic graphs

Bounds for the geometric–arithmetic index of unicyclic graphs

Atom-Bond Connectivity and Geometric Arithmetic Indices of Dendrimer Nanostars

Let G be a molecular graph. The atom-bond connectivity (ABC) and geometric-arithmetic (GA) indices of G are defined as and , where (or ) denoted the degree of the vertex u (or v), respectively. A dendrimer is a hyperbranched molecule built up from branched units called monomers. In this paper, the ABC and GA indices for two families of dendrimer nanostars are presented.

Some Topological Values of Supramolecular Chain of Different Complexes of N-Salicylidene-L-Valine

L-valine is a glycogen-type amino acid regarded among the necessary mammalian amino acids. This is an amino acid that is essential for protein synthesis. N-salicylidene-L-valine is gaining a lot of attention because of its unique structure and increased catalytic and cytotoxic activity. We explore the chain of supramolecular dialkyltin N-salicylidene-L-valine complexes 2, 3, and 4 to learn more about this structure and its features regarding topological indices. We computed the first and second Randić index, harmonic index, sum-connectivity index, atom-bond-connectivity index, geometric arithmetic index and reduced reciprocal Randić index of Supramolecular Chain of Different Complexes of N-Salicylidene-L-Valine. Furthermore, we present an analysis of such structures using specific examples, as well as a comparison of topological indices.

On eccentricity-based topological indices of line and para-line graphs of some convex polytopes

Graph theory has been studied different areas such as mathematics, information and chemistry sciences. It is about descriptors in quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) studies in the chemical network. Let G = (V(G), E(G)) be a graph without directed and multiple edges and without loops. A lot of topological indices have been defined for QSPR/QSAR studies. There are several types of these indices such as degree-based indices, eccentricity-based indices, and so on. The eccentricity-based topological indices are very important QSPR/QSAR studies, also recently these indices have been studied in many papers. In this paper, some eccentricity-based topological indices namely the connective eccentricity index xce(G), the eccentric connectivity index xc(G), the modified eccentric connectivity index xc(G), the total eccentricity index x(G), the Zagreb eccentricity indices M*1(G), M**1(G), M*2(G), the average eccentricity index avec(G), the eccentricity based geometric-arithmetic index GA4(G) and new version of ABC index such as ABC5(G) have been computed for line and para-line graphs of some convex polytopes Dn, Qn and Rn which are geometric graphs.

Mathematical aspects and topological properties of two chemical networks

<abstract><p>Graphs give a mathematical model of molecules, and thery are used extensively in chemical investigation. Strategically selections of graph invariants (formerly called "topological indices" or "molecular descriptors") are used in the mathematical modeling of the physio-chemical, pharmacologic, toxicological, and other aspects of chemical compounds. This paper describes a new technique to compute topological indices of two types of chemical networks. Our research examines the mathematical characteristics of molecular descriptors, particularly those that depend on graph degrees. We derive a compact mathematical analysis and neighborhood multiplicative topological indices for product of graphs ($ \mathcal{L} $) and tetrahedral diamond lattices ($ \Omega $). In this paper, the fifth multiplicative Zagreb index, the general fifth multiplicative Zagreb index, the fifth multiplicative hyper-Zagreb index, the fifth multiplicative product connectivity index, the fifth multiplicative sum connectivity index, the fifth multiplicative geometric-arithmetic index, the fifth multiplicative harmonic index and the fifth multiplicative redefined Zagreb index are determined. The comparison study of these topological indices is also discussed.</p></abstract>

Computation of Benzenoid Planar Octahedron Networks by Using Topological Indices

Chemical descriptors are numeric numbers that contain a basic chemical structure and describe the structure of a graph. A graph’s topological indices are linked to its chemical characteristics. Biological activity of chemical compounds can be predicted using topological indices. Numerous chemical indices have been developed in theoretical chemistry, including the Zagreb index, the Randić index, the Wiener index, and many others. In this paper, we compute the exact results for the Randić, Zagreb, Harmonic, augmented Zagreb, atom‐bond connectivity, and geometric‐arithmetic indices for the Benzenoid networks theoretically.

On arithmetic-geometric and geometric-arithmetic indices of graphs

On arithmetic-geometric and geometric-arithmetic indices of graphs

Analysis of a Productive Topological Index Correlated to Fullerenes’ Physical Properties

Fullerene is a cage-like carbon allotrope admitting a vast range of applications. Some of the important fullerenes are C54,C58,C60,C70,C74,C76,C78,C80C82,C84,C86,C90. The physical properties of fullerenes can be exhibited using the degree-based topological indices. The sum based geometric arithmetic index is significant in this manner. The sum based atomic bond connectivity, Randic, first and second Zagreb indices are well known topological indices. We have determined the regression relation between each of these indices and the sum based geometric arithmetic index. Moreover, the correlation coefficient is also calculated. Correlation is a symmetric relation, as it provides association between two variables. On the basis of regression analysis and correlation coefficient, it was found that each of this index is strongly related to the sum based geometric arithmetic index. Moreover, we have computed the regression relations concerning the physical properties depending on the sum based geometric arithmetic index. The physical properties include binding energies, Ramsauer-Townsend minima, shape resonances and heat of formation of fullerene molecules. It was concluded that the sum based GA index is the best in presenting the heat of the formation of molecules.

Computing Topological Invariants of Deep Neural Networks.

A deep neural network has multiple layers to learn more complex patterns and is built to simulate the activity of the human brain. Currently, it provides the best solutions to many problems in image recognition, speech recognition, and natural language processing. The present study deals with the topological properties of deep neural networks. The topological index is a numeric quantity associated to the connectivity of the network and is correlated to the efficiency and accuracy of the output of the network. Different degree-related topological indices such as Zagreb index, Randic index, atom-bond connectivity index, geometric-arithmetic index, forgotten index, multiple Zagreb indices, and hyper-Zagreb index of deep neural network with a finite number of hidden layers are computed in this study.

Upper bound for the geometric-arithmetic index of trees with given domination number

Given a graph G with vertex set V(G) and edge set E(G), the geometric-arithmetic index is the valueGA(G)=∑uv∈E(G)2dudvdu+dv, where du and dv denote the degrees of the vertices u,v∈V(G), respectively. In this work we present an upper bound for the geometric-arithmetic index of trees in terms of the order and the domination number, and we characterize the extremal trees for this upper bound. Finally, using a known relation between the geometric-arithmetic and arithmetic-geometric indices, we deduce a lower bound for the arithmetic-geometric index using the same parameters.

Neighborhood Versions of Geometric–Arithmetic and Atom Bond Connectivity Indices of Some Popular Graphs and Their Properties

In this article, we introduce the neighborhood versions of two classical topological indices, namely neighborhood geometric–arithmetic and neighborhood atom bond connectivity indices. We study the graph-theoretic properties of these new topological indices for some known graphs, e.g., complete graph Kn, regular graph Rn, cycle graph Cn, star graph Sn, pendant graph, and irregular graph and further establish their respective bounds. We note that the neighbourhood geometric–arithmetic index of Kn, Rn, Cn, and Sn is equal to the number of edges. The neighborhood atom bond connectivity index of an arbitrary simple graph G is strictly less than the number of edges. Our results contribute to the literature in this direction.

Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups.

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 of a finite group is a graph where non-central elements of are its vertex set, while two different elements are edge connected when they do not commute in . 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 . We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

On arithmetic–geometric eigenvalues of graphs

Abstract In this article, we are interested in characterizing graphs with three distinct arithmetic–geometric eigenvalues. We provide the bounds on the arithmetic–geometric energy of graphs. In addition, we carry out a statistical analysis of arithmetic–geometric energy and boiling point of alkanes. We observe that arithmetic–geometric energy is better correlated with a boiling point than the arithmetic–geometric index.

Certain topological indices of Basava wheel windmill graph

In this paper, we have proposed new windmill graph, that is Basava wheel windmill graph. The Basava wheel windmill graph \(W^{(m)}_{n+1}\) is the graph obtained by taking \(m\geq 2\) copies of the graph \(K_1+W_{n}\) for \(n\geq 4\) with a vertex \(K_1\) in common. Inspired by recent work on topological indices, proposed new degree-based topological indices namely, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of a graph \(G\). We have obtained first and second Zagreb index, F-index, first and second hyper-Zagreb index, harmonic index, Randi\(\acute{c}\) index, general Randi\(\acute{c}\) index, sum connectivity index, general sum connectivity index, atom-bond connectivity index, geometric-arithmetic index, Symmetric division deg index, Sombor index, SK indices, general \(SK_{\alpha}\) and \(SK^{\alpha}_1\) indices of Basava wheel windmill graph. Further, we have computed exact values of these topological indices of chloroquine, hydroxychloroquine and remdesiver.

On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider G as a group and H as a non-empty subset of G. The commuting graph C(G,H), has H as the vertex set, where h1,h2∈H are edge connected whenever h1 and h2 commute in G. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of SL(2,C). Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.