Geometric Arithmetic Research Articles

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.

Comparative Study of Certain Synthetic Polymers via Bond-Additive Invariants

Polymers, like vulcanized rubber, bakelite, and poly-methyl methacrylate (PMMA), are widely utilized as denture based materials, and their prominence has been nothing short of excellent. Recently, Ahmad et al. [Open Chemistry 17(2019): 663-670] computed bond-additive invariants (BAIs) for the molecular graph of bakelite. In the same paper, they proposed the comparative study of the aforementioned-polymers using BAIs. This paper develops molecular graphs of vulcanized rubber and PMMA to estimate M-Polynomial and the generalized Zagreb index. We derive numerous BAIs such as the first and the second Zagreb, Re-defined Zagreb, general Randić, first general Zagreb, and symmetric division degree invariants from the generalized Zagreb index. Moreover, we obtain the modified second Zagreb, inverse Randić, harmonic, inverse sum, and augmented Zagreb invariants from the M-Polynomials. Besides, we compute the atom bond connectivity, its fourth version ABC <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> , the geometric arithmetic, its fifth version GA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> , and the Sanskruti indices. Finally, we provide insight into the numerical comparison among several BAIs to establish a relation for the underlying polymers' various physicochemical properties.

On Knowledge Discovery and Representations of Molecular Structures Using Topological Indices

Abstract The main purpose of a topological index is to encode a chemical structure by a number. A topological index is a graph invariant, which decribes the topology of the graph and remains constant under a graph automorphism. Topological indices play a wide role in the study of QSAR (quantitative structure-activity relationship) and QSPR (quantitative structure-property relationship). Topological indices are implemented to judge the bioactivity of chemical compounds. In this article, we compute the ABC (atom-bond connectivity); ABC 4 (fourth version of ABC), GA (geometric arithmetic) and GA 5 (fifth version of GA) indices of some networks sheet. These networks include: octonano window sheet; equilateral triangular tetra sheet; rectangular sheet; and rectangular tetra sheet networks.

The expected values of arithmetic bond connectivity and geometric indices in random phenylene chains

Phenylenes are a class of conjugated hydrocarbons composed of a special arrangement of six- and four-membered rings. In this paper, the expected values of the atom-bond connectivity (ABC), geometric-arithmetic (GA), and Zagreb indices of this class of conjugated hydrocarbons have been determined. At the end, the comparisons with respect to the random phenylene chains among the expected values of these indices, have been determined explicitly. The graphical profiles of these indices have been shown in order to support our results.

On entropy measures of molecular graphs using topological indices

Topological indices as a molecular descriptors are important tools in (QSAR)/(QSPR) studies. The graph entropies with topological indices inspired by Shannon’s entropy concept become the information-theoretic quantities for measuring the structural information of chemical graphs and complex networks. The graph entropy measures are playing an important role in a variety of problem areas including, discrete mathematics, biology, and chemistry. Our contribution is to explore graph entropies that are based on a novel information functional, which is the number of vertices of different degree together with the number of edges between different degree vertices.In this paper, we study the chemical graph of crystal structure of titanium difluoride TiF2 and crystallographic structure of cuprite Cu2O. Also, we compute entropies of these structures by making a relation of degree based topological indices namely first redefined Zagreb index, second redefined Zagreb index, third redefined Zagreb index, fourth Atom Bond Connectivity index, fifth Geometric Arithmetic index, Sanskruti index with the help of information function.

Geometric arithmetic and mostar indices of P 2n + F Pn +1

Let G = (V, E) be a simple connected graph, where V(G) and E(G) represent the vertex set and edge set of G respectively. The vertex set V(G) associates with the atoms and the edge set E(G) associates with the bonds of the atoms in a chemical graph. For a connected graph G, the second geometric-arithmetic index GAv (G) index is denoted as , and the Mostar M (G) index of a graph G is formulated by , where nu (e) is the number of vertices which are closer to the vertex u than to vertex v of e and nv (e) is the number of vertices which are closer to vertex v than to the vertex u of e. The aim of this paper is to calculate and compare the geometric-arithmetic GAv (G) index and Mostar Mo (G) index of P 2n + FPn +1.

The Atom-Bond Connectivity and Geometric-Arithmetic Indices in Random Polyphenyl Chains

The atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are two well-studied topological indices and found to be useful tools in QSPR and QSAR investigations. We have compared the excepted values of ABC and GA indices in random spiro chains. In this paper, we further establish simple explicit formulae for the expected values of ABC and GA indices in random polyphenyl chains which are graphs of a class of unbranched polycyclic aromatic hydrocarbons and present the average values of ABC and GA indices with respect to the set of all polyphenyl chains with n hexagons. Based on these formulae, we make comparisons between the expected values of ABC and GA indices in random polyphenyl chains.

Multiplicative topological properties of graphs derived from honeycomb structure

Topological indices are numerical parameters of a molecular graph, which characterize its topology and are usually graph invariant. In quantitative structure-activity relationship/quantitative structure-property relationship study, physico-chemical properties and topological indices such as Randic, atom-bond connectivity (ABC), and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we are taking Dominating David Derived networks, produced by honeycomb structure of dimension t and obtain analytical closed results of Multiplicative topological indices and acquire exact results of degree based indices.

Computing the degree based topological indices of line graph of benzene ring embedded in P-type-surface in 2D network

A solitary number that can be utilized to describe some property of the graph of a particle is known as a topological index for that graph. There are various topological indices that have discovered a few applications in hypothetical science. In QSAR/QSPR think about, physico-concoction properties and topological indices, for example, Randic¢, atom-bond connectivity (ABC), geometric arithmetic (GA) and their edge renditions are utilized to anticipate the bioactivity of various substance mixes. In this paper, we consider the degree based molecular topological index the edge version of atom-bond connectivity index and geometric arithmetic index for line and line graph of subdivision of benzene ring embedded in P-type surface in 2D network. The diagnostic shut formulae for the edge version of atom-bond connectivity index and geometric arithmetic index of these classes of atomic structures are given

Topological Indices of mth Chain Silicate Graphs

A topological index is a numerical representation of a chemical structure, while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its numerical representation. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, and biological activity are determined by the chemical applications of graph theory. The biological activity of chemical compounds can be constructed by the help of topological indices such as atom-bond connectivity (ABC), Randić, and geometric arithmetic (GA). In this paper, Randić, atom bond connectivity (ABC), Zagreb, geometric arithmetic (GA), ABC4, and GA5 indices of the mth chain silicate S L ( m , n ) network are determined.

Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices of Generalized Bridge Molecular Graphs

Topological indices are graph invariants computed by the distance or degree of vertices of the molecular graph. In chemical graph theory, topological indices have been successfully used in describing the structures and predicting certain physicochemical properties of chemical compounds. In this paper, we propose a definition of generalized bridge molecular graphs that can model more kinds of long chain polymerization products than the bridge molecular graphs, and provide some results of the edge versions of atom-bond connectivity ( A B C e ) and geometric arithmetic ( G A e ) indices for some generalized bridge molecular graphs, which have regular, periodic and symmetrical structures. The results of this paper offer promising prospects in the applications for chemical and material engineering, especially in chemical industry research.

Computing topological polynomials of mesh-derived networks

Topological descriptors are numerical parameters of a molecular graph which characterize its molecular topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity [Formula: see text] and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. The counting polynomials are those polynomials having at exponent the extent of a property partition and coefficients the multiplicity/occurrence of the corresponding partition. All of the studied interconnection mesh networks in this paper are motivated by the molecular structure of a Sodium chloride NaCl. In this paper, Omega, Sadhana and PI polynomials are computed for mesh-derived networks. These polynomials were proposed on the ground of quasi-orthogonal cut edge strips in polycyclic graphs. These polynomials count equidistant and non-equidistant edges in graphs. Moreover, the analytical closed formulas of these polynomials for mesh-derived networks are computed for the first time.

On Degree-Based Topological Indices of Symmetric Chemical Structures

A Topological index also known as 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 which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study HDCN1(m,n) and HDCN2(m,n) of dimension m , n and derive analytical closed results of general Randić index R α ( G ) for different values of α . We also compute the general first Zagreb, ABC, GA, A B C 4 and G A 5 indices for these Hex derived cage networks for the first time and give closed formulas of these degree-based indices.

Computing multiple ABC index and multiple GA index of some grid graphs

AbstractTopological indices are the atomic descriptors that portray the structures of chemical compounds and they help us to anticipate certain physico-compound properties like boiling point, enthalpy of vaporization and steadiness. The atom bond connectivity (ABC) index and geometric arithmetic (GA) index are topological indices which are defined as$ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}}$and$GA(G)=\sum_{uv\in E(G)}\frac{2\sqrt{d_ud_v}}{d_u+d_v}$, respectively, whereduis the degree of the vertexu. The aim of this paper is to introduced the new versions ofABCindex andGAindex namely multiple atom bond connectivity (ABC) index and multiple geometric arithmetic (GA) index. As an application, we have computed these newly defined indices for the octagonal grid$O_p^q$, the hexagonal gridH(p,q) and the square gridGp, q. Also, we compared these results obtained with the ones by other indices like theABC4index and theGA5index.

Edge Version Molecular Descriptors of Tetrameric 1, 3 Adamantane

Molecular descriptors (Topological indices) are the numerical invariants of a molecular graph which distinguish its topology. In this article, we compute edge version of topological indices such as Zagreb index, Atom bond connectivity index, Fourth atom bond connectivity index, Geometric Arithmetic index and Fifth Geometric Arithmetic index of tetrameric 1,3 adamantane.

About Atom Bond Connectivity and Geometric-Arithmetic Indices of Special Chemical Molecular and Nanotubes

Among topological descriptors connectivity indices are very important and they have a prominent role in chemistry. Two useful of them are the geometric-arithmetic (GA) and atom-bond connectivity ( ABC ) indices and are defined as \(GA(G)=\sum\limits _{uv\in E(G)}\frac{2\sqrt{d_{u} d_{v} } }{d_{u} +d_{v} }\) and \(ABC(G)=\sum\limits _{e=uv\in E(G)}\sqrt{\frac{d_{u} +d_{v} -2}{d_{u} d_{v} } }\), in which \(d_u\) and \(d_v\) are the degrees of the vertices \(u\) and \(v\), respectively. n this paper we compute these connectivity topological indices for a special chemical molecular graph ``$Cas(C)$-$CaR(C)[m,n,p]$ Nanotubes Junction'' are given. The $Cas(C)$-$CaR(C)[m,n,p]$ Nanotubes Junction is a new nano-structure that was defined by M.V. Diudea, on based the new graph operations (Leapfrog Le and Capra Ca) on the cycle graph \(C_n\).

On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si2C3-I[p,q] and Si2C3-II[p,q

The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topological indices help us to predict many physico-chemical properties of the concerned molecular compound. In this article, we compute Generalized first and multiplicative Zagreb indices, the multiplicative version of the atomic bond connectivity index, and the Generalized multiplicative Geometric Arithmetic index for silicon-carbon Si2C3−I[p,q] and Si2C3−II[p,q] second.

The Cartesian Product and Join Graphs on Edge-Version Atom-Bond Connectivity and Geometric Arithmetic Indices.

The Cartesian product and join are two classical operations in graphs. Let be the degree of a vertex in line graph of a graph . The edge versions of atom-bond connectivity () and geometric arithmetic () indices of G are defined as and , respectively. In this paper, and indices for certain Cartesian product graphs (such as , and ) are obtained. In addition, and indices of certain join graphs (such as , , and ) are deduced. Our results enrich and revise some known results.

On ve-degree molecular topological properties of silicate and oxygen networks

Silicate based inorganic materials are important for the synthesis of a new inorganic molecules in which the studies for ultrahigh proton conductivity and catalysis. Quantitative structure-property and structure-activity relationships of the silicate oxygen networks necessitate expressions for the molecular topological features of these networks. In QSPR/QSAR studies, physicochemical characteristics and molecular topological indices such as atom-bond connectivity (ABC), geometric-arithmetic (GA), harmonic (H) and sum-connectivity (χ) indices are used to model the physicochemical properties of chemical compounds and networks. These topological indices are based on the degrees of the vertices (atoms) of a connected graph. Recently, two novel degree concepts have been defined in graph theory; ev-degrees and ve-degrees. In this study by using the ve-degree concept, we define ve-degree atom-bond connectivity (ve-ABC), ve-degree geometric-arithmetic (ve-GA), ve-degree harmonic (ve-H) and ve-degree sum-connectivity (ve-χ) indices as parallel to their corresponding classical degree versions. We show that the ve-degree sum-connectivity index give better correlation than Wiener, Zagreb and Randic indices to predict the acentric factor of octanes. Also, we compute the ve-degree topological indices for some silicate oxygen netwoks such as dominating oxide network (DOX), regular triangulene oxide network (RTOX), dominating silicate network (DSL) and derive analytical closed formulae of these networks.

On the bounds of degree-based topological indices of the Cartesian product of F-sum of connected graphs

Topological indices are the mathematical tools that correlate the chemical structure with various physical properties, chemical reactivity or biological activity numerically. A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. In QSAR/QSPR study, a prediction about the bioactivity of chemical compounds is made on the basis of physico-chemical properties and topological indices such as Zagreb, Randić and multiple Zagreb indices. In this paper, we determine the lower and upper bounds of Zagreb indices, the atom-bond connectivity (ABC) index, multiple Zagreb indices, the geometric-arithmetic (GA) index, the forgotten topological index and the Narumi-Katayama index for the Cartesian product of F-sum of connected graphs by using combinatorial inequalities.