Analysis of COVID-19 by Means of Graph Theory

As a powerful displaying, investigation and computational device, graph theory is widely used in biological mathematics to deal with various biology problems. In the field of microbiology, graphs can communicate the sub-atomic structure. Where cell, quality or protein can be indicated as a vertex, and the associate component can be viewed as an edge. Thusly, the biological activity characteristic can be measured via topological index computing in the comparing graphs. In this article, we for the most part concentrate some topological lists for the Corona virus graph. At first,we give a general type of M-polynomial. From the M-polynomial, we recoup some well-known degree-based topological lists, for example, First and Second Zagreb Indices, Second Modified Zagreb Index, Randic´ Index, General Randic´ Index, Symmetric Division Index, Harmonic Index, Inverse Sum Index, Augmented Zagreb Index. Our results are extensions of many existing results.

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Topology-Based Analysis of OTIS (Swapped) Networks OKn and OPn

Journal of Chemistry ◽

10.1155/2019/4291943 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11

Author(s):

Hai-Xia Li ◽

Sarfaraz Ahmad ◽

Iftikhar Ahmad

Keyword(s):

Topological Index ◽

Molecular Descriptor ◽

Randić Index ◽

Zagreb Index ◽

Randic Index ◽

Chemical Graph ◽

Zagreb Indices ◽

Second Zagreb Index ◽

Chemical Graph Theory ◽

Augmented Zagreb Index

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In this paper, M-polynomial OKn and OPn networks are computed. The M-polynomial is rich in information about degree-based topological indices. By applying the basic rules of calculus on M-polynomials, the first and second Zagreb indices, modified second Zagreb index, general Randić index, inverse Randić index, symmetric division index, harmonic index, inverse sum index, and augmented Zagreb index are recovered.

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M-Polynomials and Degree-Based Topological Indices of the Molecule Copper(I) Oxide

Journal of Chemistry ◽

10.1155/2021/6679819 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Faryal Chaudhry ◽

Iqra Shoukat ◽

Deeba Afzal ◽

Choonkil Park ◽

Murat Cancan ◽

...

Keyword(s):

Topological Indices ◽

Randić Index ◽

Zagreb Index ◽

Harmonic Index ◽

Randic Index ◽

Zagreb Indices ◽

Symmetric Division ◽

Augmented Zagreb Index ◽

Physical And Chemical ◽

Modified Zagreb Index

Topological indices are numerical parameters used to study the physical and chemical residences of compounds. Degree-based topological indices have been studied extensively and can be correlated with many properties of the understudy compounds. In the factors of degree-based topological indices, M-polynomial played an important role. In this paper, we derived closed formulas for some well-known degree-based topological indices like first and second Zagreb indices, the modified Zagreb index, the symmetric division index, the harmonic index, the Randić index and inverse Randić index, and the augmented Zagreb index using calculus.

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On Leap Reduced Reciprocal Randic and Leap Reduced Second Zagreb Indices of Some Graphs

Scientific Inquiry and Review ◽

10.32350/sir.32.04 ◽

2019 ◽

Vol 3 (2) ◽

pp. 27-35

Author(s):

Fazal Dayan ◽

Muhammad Javaid ◽

Muhammad Aziz ur Rehman

Keyword(s):

Topological Indices ◽

Randić Index ◽

Zagreb Index ◽

Randic Index ◽

Zagreb Indices ◽

Second Zagreb Index ◽

Second Degree

Naji et al. introduced the leap Zagreb indices of a graph in 2017 which are new distance-degree-based topological indices conceived depending on the second degree of vertices. In this paper, we have defined the first and second leap reduced reciprocal Randic index and leap reduced second Zagreb index for selected wheel related graphs.

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On Computation Degree-Based Topological Descriptors for Planar Octahedron Networks

Journal of Mathematics ◽

10.1155/2021/4880092 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Wang Zhen ◽

Parvez Ali ◽

Haidar Ali ◽

Ghulam Dustigeer ◽

Jia-Bao Liu

Keyword(s):

Graph Theory ◽

Chemical Compound ◽

Topological Index ◽

Molecular Graph ◽

Topological Descriptors ◽

Zagreb Index ◽

Chemical Graph ◽

Symmetric Division ◽

Chemical Graph Theory ◽

Augmented Zagreb Index

A molecular graph is used to represent a chemical molecule in chemical graph theory, which is a branch of graph theory. A graph is considered to be linked if there is at least one link between its vertices. A topological index is a number that describes a graph’s topology. Cheminformatics is a relatively young discipline that brings together the field of sciences. Cheminformatics helps in establishing QSAR and QSPR models to find the characteristics of the chemical compound. We compute the first and second modified K-Banhatti indices, harmonic K-Banhatti index, symmetric division index, augmented Zagreb index, and inverse sum index and also provide the numerical results.

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General Randić index of unicyclic graphs with given girth and diameter

Asian-European Journal of Mathematics ◽

10.1142/s1793557122501650 ◽

2021 ◽

Author(s):

Tomáš Vetrík

Keyword(s):

Lower Bounds ◽

Randić Index ◽

Zagreb Index ◽

Unicyclic Graphs ◽

Randic Index ◽

General Randić Index ◽

General Randic Index ◽

Modified Zagreb Index

We study the general Randić index of a graph [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] is the edge set of [Formula: see text] and [Formula: see text] and [Formula: see text] are the degrees of vertices [Formula: see text] and [Formula: see text], respectively. For [Formula: see text], we present lower bounds on the general Randić index for unicyclic graphs of given diameter and girth, and unicyclic graphs of given diameter. Lower bounds on the classical Randić index and the second modified Zagreb index are corollaries of our results on the general Randić index.

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Some Topological Invariants of Graphs Associated with the Group of Symmetries

Journal of Chemistry ◽

10.1155/2020/6289518 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Chang-Cheng Wei ◽

Muhammad Salman ◽

Usman Ali ◽

Masood Ur Rehman ◽

Muhammad Aqeel Ahmad Khan ◽

...

Keyword(s):

Biological Activity ◽

Topological Index ◽

Chemical Reactivity ◽

Wiener Index ◽

Structural Features ◽

Randić Index ◽

Harary Index ◽

Randic Index ◽

Chemical Structures ◽

Connectivity Indices

A topological index is a quantity that is somehow calculated from a graph (molecular structure), which reflects relevant structural features of the underlying molecule. It is, in fact, a numerical value associated with the chemical constitution for the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. A large number of properties like physicochemical properties, thermodynamic properties, chemical activity, and biological activity can be determined with the help of various topological indices such as atom-bond connectivity indices, Randić index, and geometric arithmetic indices. In this paper, we investigate topological properties of two graphs (commuting and noncommuting) associated with an algebraic structure by determining their Randić index, geometric arithmetic indices, atomic bond connectivity indices, harmonic index, Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, and Harary index.

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On multiplicative degree based topological indices for planar octahedron networks

Main Group Metal Chemistry ◽

10.1515/mgmc-2020-0026 ◽

2020 ◽

Vol 43 (1) ◽

pp. 219-228

Author(s):

Ghulam Dustigeer ◽

Haidar Ali ◽

Muhammad Imran Khan ◽

Yu-Ming Chu

Keyword(s):

Graph Theory ◽

Information Science ◽

Biological Activities ◽

Molecular Graph ◽

Triangular Prism ◽

Structure Property ◽

Zagreb Index ◽

Zagreb Indices ◽

Chemical Graph Theory ◽

Analytical Formulas

AbstractChemical graph theory is a branch of graph theory in which a chemical compound is presented with a simple graph called a molecular graph. There are atomic bonds in the chemistry of the chemical atomic graph and edges. The graph is connected when there is at least one connection between its vertices. The number that describes the topology of the graph is called the topological index. Cheminformatics is a new subject which is a combination of chemistry, mathematics and information science. It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. We evaluated the second multiplicative Zagreb index, first and second universal Zagreb indices, first and second hyper Zagreb indices, sum and product connectivity indices for the planar octahedron network, triangular prism network, hex planar octahedron network, and give these indices closed analytical formulas.

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Multiplicative Zagreb indices of cacti

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500403 ◽

2016 ◽

Vol 08 (03) ◽

pp. 1650040 ◽

Cited By ~ 6

Author(s):

Shaohui Wang ◽

Bing Wei

Keyword(s):

Graph Theory ◽

Lower Bounds ◽

Connected Graph ◽

Upper And Lower Bounds ◽

Extremal Graphs ◽

Zagreb Index ◽

Zagreb Indices ◽

Material Chemistry ◽

Cactus Graph ◽

Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

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Bounds on General Randić Index for F-Sum Graphs

Journal of Mathematics ◽

10.1155/2020/9129365 ◽

2020 ◽

Vol 2020 ◽

pp. 1-17

Author(s):

Xu Li ◽

Maqsood Ahmad ◽

Muhammad Javaid ◽

Muhammad Saeed ◽

Jia-Bao Liu

Keyword(s):

Molecular Graph ◽

Randić Index ◽

Lower And Upper Bounds ◽

First Zagreb Index ◽

Zagreb Index ◽

Randic Index ◽

Structure Activity ◽

General Randić Index ◽

General Randic Index ◽

The First Zagreb Index

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

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First General Zagreb Index of Generalized F-sum Graphs

Discrete Dynamics in Nature and Society ◽

10.1155/2020/2954975 ◽

2020 ◽

Vol 2020 ◽

pp. 1-16 ◽

Cited By ~ 1

Author(s):

H. M. Awais ◽

Muhammad Javaid ◽

Akbar Ali

Keyword(s):

Real Number ◽

Cartesian Product ◽

Randić Index ◽

Zeroth Order ◽

Zagreb Index ◽

Randic Index

The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph G can be defined as M γ G = ∑ u v ∈ E G d G γ − 1 u + d G γ − 1 v , where γ is a real number. As M γ G is equal to the order and size of G when γ = 0 and γ = 1 , respectively, γ is usually assumed to be different from 0 to 1. In this paper, for every integer γ ≥ 2 , the FGZ index M γ is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. The obtained results can be considered as the generalizations of the results appeared in (IEEE Access; 7 (2019) 47494–47502) and (IEEE Access 7 (2019) 105479–105488).

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