Hyper Zagreb Index Research Articles

On the Hyper Zagreb Index of Trees with a Specified Degree of Vertices

Topological indices are the numerical descriptors that correspond to some certain physicochemical properties of a chemical compound such as the boiling point, acentric factor, enthalpy of vaporisation, heat of fusion, etc. Among these topological indices, the Hyper Zagreb index, is the most effectively used topological index to predict the acentric factor of some octane isomers. In the current work, we investigate the extremal values of the Hyper Zagreb index for some classes of trees.

On expected values of some degree based topological descriptors of random Phenylene chains

Phenylenes belong to a special class of conjugated hydrocarbons composed of a special arrangement of hexagons and squares. In the phenylene structure, every square is adjacent to a pair of hexagons, and any two hexagons are not adjacent. If in the structure of phenylene only two squares are adjacent to every hexagon, then it is called a phenylene chain. In this paper, we compute the expected values of the first and second Gourava index, the first and second redefined Zagreb index, and the Hyper-Zagreb index of random phenylene chains. Furthermore, a comparison between the expected values of the computed indices among the classes of random phenylene chains has been investigated.

Novel Degree-Based Topological Descriptors of Fenofibrate Using M-Polynomial

Chemical graph theory is currently expanding the use of topological indices to numerically encode chemical structure. The prediction of the characteristics provided by the chemical structure of the molecule is a key feature of these topological indices. The concepts from graph theory are presented in a brief discussion of one of its many applications to chemistry, namely, the use of topological indices in quantitative structure-activity relationship (QSAR) studies and quantitative structure-property relationship (QSPR) studies. This study uses the M-polynomial approach, a newly discovered technique, to determine the topological indices of the medication fenofibrate. With the use of degree-based topological indices, we additionally construct a few novel degree based topological descriptors of fenofibrate structure using M-polynomial. When using M-polynomials in place of degree-based indices, the computation of the topological indices can be completed relatively quickly. The topological indices are also plotted. Using M-polynomial, we compute novel formulas for the modified first Zagreb index, modified second Zagreb index, first and second hyper Zagreb indices, SK index, S K 1 index, S K 2 index, modified Albertson index, redefined first Zagreb index, and degree-based topological indices.

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.

The Hyper-Zagreb Index of Trees and Unicyclic Graphs

The Hyper-Zagreb Index of Trees and Unicyclic Graphs

Second leap hyper-Zagreb coindex of certain benzenoid structures and their polynomials

Topological indices are well known for their importance in determining the characteristics of chemical compounds. On the other side, benzenoid hydrocarbons are essential to the food and chemical industries and our environment. In this paper, the second leap hyper-Zagreb coindex and ψ2-polynomial for certain classes of benzenoid structures such as zigzag benzenoid Zp, rhombic benzenoid Rp, multiple segment linear hexagon chains L[p,q], and hammer-like benzenoid Hp have been computed. Furthermore, numerical comparisons between the second leap hyper-Zagreb index and its coindex have been performed using Matlab. Also, correlation between the second leap hyper-Zagreb index and its coindex with physico-chemical properties of some polycyclic aromatic hydrocarbons have been predicted.

On multiplicative universal Zagreb and its subsequent indices of C4C8 carbon nanostructures

Mathematical chemistry studies the chemical structure of molecules. Topological indices are numerical values which associates the chemical structure with the physical and chemical properties. Multiplicative Universal Zagreb indices are generalized degree-based topological indices which gave rise to several indices like first and second multiplicative, Zagreb and hyper-Zagreb indices, multiplicative sum and product conductivity indices. This research is designed to study the first and second multiplicative Universal Zagreb indices of carbon nanostructures (carbon nanosheet, nanotube and nanotorus) of the same chemical formula [Formula: see text] We also depict the values of first and second multiplicative, Zagreb and hyper-Zagreb indices as well as multiplicative sum and product connectivity indices for the mentioned structures. The graphical comparison for each of the multiplicative Zagreb indices is presented for all the carbon nanostructures with the same chemical formula [Formula: see text]

Generalized Quasi Trees with Respect to Degree Based Topological Indices and Their Applications to COVID-19 Drugs

The l-generalized quasi tree is a graph G for which we can find W⊂V(G) with |W|=l such that G−W is a tree but for an arbitrary Y⊂V(G) with |Y|<l, G−Y is not a tree. In this paper, inequalities with respect to zeroth-order Randić and hyper-Zagreb indices are studied in the class of l-generalized quasi trees. The corresponding extremal graphs corresponding to these indices in the class of l-generalized quasi trees are also obtained. In addition, we carry QSPR analysis of COVID-19 drugs with zeroth-order Randić and hyper-Zagreb indices (energy).

Algebraic Properties for Molecular Structure of燤agnesium營odide

As an inorganic chemical, magnesium iodide has a significant crystalline structure. It is a complex and multi-functional substance that has the potential to be used in a wide range of medical advancements. Molecular graph theory, on the other hand, provides a sufficient and cost-effective method of investigating chemical structures and networks. M-polynomial is a relatively new method for studying chemical networks and structures in molecular graph theory. It displays numerical descriptors in algebraic form and highlights molecular features in the form of a polynomial function. We present a polynomials display of magnesium iodide structure and calculate several M-polynomials in this paper, particularly the M-polynomials of the augmented Zagreb index, inverse sum index, hyper Zagreb index and for the symmetric division index.

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>

An investigation of edge F-index on fuzzy graphs and application in molecular chemistry

The molecular descriptors are a useful tool in the spectral graph, molecular chemistry and several fields of chemistry and mathematics. The edge F-index is proposed for fuzzy graphs (FGs) here. Bounds of this index are calculated for FGs. The FG has been investigated for a given set of vertices as having maximum edge F-index. Some relations of this index with the second Zagreb index and hyper-Zagreb index are established. For an isomorphic FGs, it is shown that the value of this index is the same. Bounds of this index for some FG operations are determined. Also, an application of the index in mathematical chemistry is studied. For this, 18 octane isomers and 67 alkanes are considered and analyzed the correlation between this index with some properties of the octane isomers and alkanes. From the correlation coefficient value, we have obtained this index is highly correlated with enthalpy of vaporization, standard enthalpy of vaporization, entropy, acentric factor and heat of vaporization and less correlated with heat capacity for octane isomers. Also, this index is correlated with critical pressure, critical temperature, molar refraction, molar volume and boiling point and is less correlated with surface tension and heat of vaporization for alkanes. But, this index is inadequate to determine the melting point of alkanes.

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.

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.

A Statistical Comparison between Zagreb indices for correlation with toxicity predictions of natural products

Graph theory had wide applications in developing in silico tools and it is widely used to calculate topological indices to establish structural activity relations of chemicals/compounds. However, usage of Zagreb indices with respect to natural compounds activity/toxicity prediction needs more attention. Many available online tools are using atom bond connectivity index (ABC Index), first and second Zagreb indices. The usage of the Hyper Zagreb index is very rare and using natural compounds is neglected. In this context, three types of Zagreb indices (first Zagreb index, second Zagreb index and hyper Zagreb index) were calculated to the selected chemical compounds of natural products and the relation between these indices and cytotoxicity of natural compounds were established. We have selected IC50 Values of the selected natural compounds in Hela cell lines as an index for cytotoxicity from the literature. The correlation of Zagreb indices and activity was performed using the R program, and we reached the conclusion that all indices correlate with the cytotoxicity of the studied compounds. This study acts as evidence to prove that, hyper Zagreb index correlates more with the cytotoxicity/activity of the studied natural compounds. Further studies using other Machine Learning tools to verify these findings will establish the importance of the hyper Zagreb index as one method to predict the toxicity of natural compounds.

Computing some Novel Closed Neighborhood Degree-Based Topological Indices of Graphene Structures

Chemical graph theory (CGT) is a crucial part of mathematical chemistry, which characterizes the chemical compounds as molecular graphs using graph-theoretical tools. By IUPAC terminology, a numerical parameter that represents the topology of a molecular graph is known as a topological index (TI). In this scenario, our research interest is to employ newly defined seven TIs, namely closed neighborhood first Zagreb index (CM1(G)), modified closed neighborhood first Zagreb index (CM1*(G)), closed neighborhood second Zagreb index (CM2(G)), closed neighborhood Forgotten index (CF(G)), modified closed neighborhood Forgotten index (CF*(G)), closed neighborhood first hyper Zagreb index (CHM1(G)) and closed neighborhood second hyper Zagreb index (CHM2(G)) as well as explore them mathematically. Without applying quantum mechanics, these TIs could obtain the maximum information behind the graphene structures.

Topological Invariants for the General Structure of Grape Seed Proanthocyanidins

In this paper, we formulate the degree-based topological indices such as first Zagreb index, second Zagreb index, third Zagreb index, atom bond connectivity index, geometric-arithmetic (GA) index, general sum connectivity index, hyper-Zagreb index, augmented Zagreb index, sum connectivity index and general Randic index Rλ (℘), for λ = © −1, 1, − 1 2 , 1 2 ª for grape seed Proanthocyanidins network. By using above-mentioned topological indices, this study provides us different outcomes.

Computing Topological Indices and Polynomials of the Rhenium Trioxide

In the study of mathematical chemistry and chemical graph theory, a topological index, also known as a connectivity index, is the arithmetical framework of a graph that specifies its topology and also graph invariant. These topological indices are used to model quantitative structure relationships (QSARs), which are connections between the work of biological or other molecular structures and the chemical structures. This study computed the first, second, and Hyper Zagreb indices, as well as Zagreb polynomials, Redefined Zagreb indices, Randic index, ABC index, and GA index of chemical structure of Rhenium Trioxide.

Comparative Study of Generalized Sum Graphs via Degree‐Based Topological Indices

In theoretical chemistry, topological indices (TIs) have important role to predict various physical and structural properties of the study under molecular graphs. Among all topological indices, Zagreb‐type indices have been used more effectively in the chemical literature. In this paper, we have computed first Zagreb, second Zagreb, forgotten, and hyper Zagreb indices of the generalized Q‐sum graph in the form of different TIs of its basic graphs, where α ≥ 1 is a positive integer. This family of graphs is obtained by the lexicographic product of the graph Qα(H1) and H2, where Qα(H1) is constructed with the help of the generalized line superposition operation Qα on H1. As a conclusion, we also checked the correlation between predefined graph H1[H2] under the operation of lexicographic product H1 and H2 with newly defined generalized Q‐sum graphs using linear regression models of various degree‐based TIs.

Computing Topological Descriptors and Polynomials of Certain 2D Chemical Structures

In the fields of 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 an analytical framework of a graph which portray its topology and are mostly equal graphs. Topological indices (TIs) are numeral quantities that are used to foresee the natural correlation among the physicochemical properties of the chemical compounds in their fundamental network. TIs show an essential role in the theoretical abstract and environmental chemistry and pharmacology. In this paper, we compute many latest developed degree-based TIs. An analogy among the computed different versions of the TIs with the help of the numerical values and their graphs is also included .In this article, we compute the first Zagreb index, second Zagreb index, hyper Zagreb index, ABC Index, GA Index, and first Zagreb polynomial and second Zagreb polynomial of chemical graphs polythiophene, nylon 6,6, and the backbone structure of DNA.

The second Hyper-Zagreb index of graph operations

A graph can be recognized by numeric number, polynomial or matrix which represent the whole graph. Topological index is a numerical descriptor of a molecule, based on a certain topological feature of the corresponding molecular graph, it is found that there is a strong correlation between the properties of chemical compounds and their molecular structure. Zagreb indices are numeric numbers related to graphs. In this study, the second Hyper-Zagreb index for some special graphs, and graph operations has been computed, that have been applied to compute the second Hyper-Zagreb index for Nano-tube and Nano-torus.