scholarly journals First reformulated Zagreb indices of some classes of graphs

2018 ◽  
Vol 9 (2) ◽  
pp. 134-144 ◽  
Author(s):  
V. Kaladevi ◽  
R. Murugesan ◽  
K. Pattabiraman
Keyword(s):  
Quantitative Structure Activity Relationship ◽  
Vital Role ◽  
Pictorial Representation ◽  
Quantitative Structure ◽  
Structure Property ◽  
Zagreb Index ◽  
Qsar Studies ◽  
Zagreb Indices ◽  
Subdivision Graph ◽  
Parallel Graph

A topological index of a graph is a parameter related to the graph; it does not depend on labeling or pictorial representation of the graph. Graph operations plays a vital role to analyze the structure and properties of a large graph which is derived from the smaller graphs. The Zagreb indices are the important topological indices found to have the applications in Quantitative Structure Property Relationship(QSPR) and Quantitative Structure Activity Relationship(QSAR) studies as well. There are various study of different versions of Zagreb indices. One of the most important Zagreb indices is the reformulated Zagreb index which is used in QSPR study. In this paper, we obtain the first reformulated Zagreb indices of some derived graphs such as double graph, extended double graph, thorn graph, subdivision vertex corona graph, subdivision graph and triangle parallel graph. In addition, we compute the first reformulated Zagreb indices of two important transformation graphs such as the generalized transformation graph and generalized Mycielskian graph.

scholarly journals The Second Hyper-Zagreb Index of Complement Graphs and Its Applications of Some Nano Structures

2021 ◽  
pp. 54-75
Author(s):  
Mohammed Alsharafi ◽  
Yusuf Zeren ◽  
Abdu Alameri
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Quantitative Structure Activity Relationship ◽  
Quantitative Structure ◽  
Structure Property ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Strong Product ◽  
Chemical Graph Theory ◽  
Mathematical Operations

In chemical graph theory, a topological descriptor is a numerical quantity that is based on the chemical structure of underlying chemical compound. Topological indices play an important role in chemical graph theory especially in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR). In this paper, we present explicit formulae for some basic mathematical operations for the second hyper-Zagreb index of complement graph containing the join G1 + G2, tensor product G1 \(\otimes\) G2, Cartesian product G1 x G2, composition G1 \(\circ\) G2, strong product G1 * G2, disjunction G1 V G2 and symmetric difference G1 \(\oplus\) G2. Moreover, we studied the second hyper-Zagreb index for some certain important physicochemical structures such as molecular complement graphs of V-Phenylenic Nanotube V PHX[q, p], V-Phenylenic Nanotorus V PHY [m, n] and Titania Nanotubes TiO2.

scholarly journals Multiplicative Zagreb Indices of Molecular Graphs

2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Xiujun Zhang ◽  
H. M. Awais ◽  
M. Javaid ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Mathematical Modeling ◽  
Vital Role ◽  
Molecular Structures ◽  
Quantitative Structure ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Molecular Graphs ◽  
Structure Properties

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

scholarly journals Bounds for the Topological Indices of ℘ graph

2021 ◽  
Vol 14 (2) ◽  
pp. 340-350
Author(s):  
Muddalapuram Manjunath ◽  
V. Lokesha ◽  
. Suvarna ◽  
Sushmitha Jain
Keyword(s):  
Finite Graph ◽  
Quantitative Structure Activity Relationship ◽  
Topological Indices ◽  
Quantitative Structure ◽  
Structure Property ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Chemical Structures ◽  
Structure Property Relationship ◽  
Atom Bond

Topological indices are mathematical measure which correlates to the chemical structures of any simple finite graph. These are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR). In this paper, we define operator graph namely, ℘ graph and structured properties. Also, establish the lower and upper bounds for few topological indices namely, Inverse sum indeg index, Geometric-Arithmetic index, Atom-bond connectivity index, first zagreb index and first reformulated Zagreb index of ℘-graph.

scholarly journals Some Algebraic Polynomials and Topological Indices of Octagonal Network

2016 ◽  
Author(s):  
Young Chel Kwun ◽  
Waqas Nazeer ◽  
Mobeen Munir ◽  
Shin Min Kang
Keyword(s):  
Graphical Representation ◽  
Topological Indices ◽  
Molecular Structures ◽  
Quantitative Structure ◽  
Structure Property ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Structure Activity ◽  
Activity Structure ◽  
Augmented Zagreb Index

M-polynomial of different molecular structures helps to calculate many topological indices. A topological index of graph G is a numerical parameter related to G which characterizes its molecular topology and is usually graph invariant. In the field of quantitative structure-activity (QSAR) quantitative structure-activity structure-property (QSPR) research, theoretical properties of the chemical compounds and their molecular topological indices such as the Zagreb indices, Randic index, Symmetric division index, Harmonic index, Inverse sum index, Augmented Zagreb index, multiple Zagreb indices etc. are correlated. In this report, we compute closed forms of M-polynomial, first Zagreb polynomial and second Zagreb polynomial of Octagonal network. From the M-polynomial we recover some degree-based topological indices for Octagonal network. Moreover, we give a graphical representation of our results.

On Certain Counting Polynomial of Titanium Dioxide Nanotubes

2019 ◽  
Vol 9 (2) ◽  
pp. 240-243 ◽  
Author(s):  
S. Prabhu ◽  
M. Arulperumjothi ◽  
G. Murugan ◽  
V.M. Dhinesh ◽  
J.P. Kumar
Keyword(s):  
Biological Activities ◽  
Molecular Graph ◽  
Quantitative Structure Activity Relationship ◽  
Vital Role ◽  
Quantitative Structure ◽  
Structure Property ◽  
Covalent Bonds ◽  
Mathematical Chemistry ◽  
Topological Description ◽  
Counting Polynomial

Background: In 1936, Polya introduced the concept of a counting polynomial in chemistry. However, the subject established little attention from chemists for some decades even though the spectra of the characteristic polynomial of graphs were considered extensively by numerical means in order to obtain the molecular orbitals of unsaturated hydrocarbons. Counting polynomial is a sequence representation of a topological stuff so that the exponents precise the magnitude of its partitions while the coefficients are correlated to the occurrence of these partitions. Counting polynomials play a vital role in topological description of bipartite structures as well as counts of equidistant and non-equidistant edges in graphs. Omega, Sadhana, PI polynomials are wide examples of counting polynomials. Methods: Mathematical chemistry is a division of abstract chemistry in which we debate and forecast the chemical structure by using mathematical models. Chemical graph theory is a subdivision of mathematical chemistry in which the structure of a chemical compound can be embodied by a labelled graph whose vertices are atoms and edges are covalent bonds between the atoms. We use graph theoretic technique in finding the counting polynomials of TiO2 nanotubes. : Let ! be the molecular graph of TiO2. Then (!, !) = !!10!!+8!−2!−2 + (2! +1) !10!!+8!−2! + 2(! + 1)10!!+8!−2 Results: In this paper, the omega, Sadhana and PI counting polynomials are studied. These polynomials are useful in determining the omega, Sadhana and PI topological indices which play an important role in studies of Quantitative structure-activity relationship (QSAR) and Quantitative structure-property relationship (QSPR) which are used to predict the biological activities and properties of chemical compounds. Conclusion: These counting polynomials play an important role in topological description of bipartite structures as well as counts equidistance and non-equidistance edges in graphs. Computing distancecounting polynomial is under investigation.

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

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1026 ◽  
Author(s):  
Juan L. G. Guirao ◽  
Muhammad Imran ◽  
Muhammad Kamran Siddiqui ◽  
Shehnaz Akhter
Keyword(s):  
Biological Activities ◽  
Topological Descriptors ◽  
Quantitative Structure ◽  
Structure Property ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Structure Property Relationships ◽  
Quantitative Structure Property Relationships ◽  
Graph Operation ◽  
Atom Bond

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.

scholarly journals Comparison of Irregularity Indices of Several Dendrimers Structures

Processes ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 662 ◽  
Author(s):  
Dongming Zhao ◽  
Zahid Iqbal ◽  
Rida Irfan ◽  
Muhammad Anwar Chaudhry ◽  
Muhammad Ishaq ◽  
...  
Keyword(s):  
Chemical Properties ◽  
Quantitative Structure Activity Relationship ◽  
Quantitative Structure ◽  
Structure Property ◽  
Qsar Studies ◽  
Irregularity Index ◽  
Boiling Points ◽  
Vertex Degrees ◽  
Physical And Chemical

Irregularity indices are usually used for quantitative characterization of the topological structures of non-regular graphs. In numerous problems and applications, especially in the fields of chemistry and material engineering, it is useful to be aware of the irregularity of a molecular structure. Furthermore, the evaluation of the irregularity of graphs is valuable not only for quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies but also for various physical and chemical properties, including entropy, enthalpy of vaporization, melting and boiling points, resistance, and toxicity. In this paper, we will restrict our attention to the computation and comparison of the irregularity measures of different classes of dendrimers. The four irregularity indices which we are going to investigate are σ irregularity index, the irregularity index by Albertson, the variance of vertex degrees, and the total irregularity index.

Computational Approaches for the Design of Mosquito Repellent Chemicals

2020 ◽  
Vol 27 (1) ◽  
pp. 32-41 ◽  
Author(s):  
Subhash C. Basak ◽  
Apurba K. Bhattacharjee
Keyword(s):  
Computational Design ◽  
Quantitative Structure Activity Relationship ◽  
Computer Assisted ◽  
Quantitative Structure ◽  
Mosquito Repellent ◽  
Computational Approaches ◽  
Qsar Studies ◽  
Computer Assisted Design ◽  
Structure Activity ◽  
Mosquito Repellents

Background: In view of many current mosquito-borne diseases there is a need for the design of novel repellents. Objective: The objective of this article is to review the results of the researches carried out by the authors in the computer-assisted design of novel mosquito repellents. Methods: Two methods in the computational design of repellents have been discussed: a) Quantitative Structure Activity Relationship (QSAR) studies from a set of repellents structurally related to DEET using computed mathematical descriptors, and b) Pharmacophore based modeling for design and discovery of novel repellent compounds including virtual screening of compound databases and synthesis of novel analogues. Results: Effective QSARs could be developed using mathematical structural descriptors. The pharmacophore based method is an effective tool for the discovery of new repellent molecules. Conclusion: Results reviewed in this article show that both QSAR and pharmacophore based methods can be used to design novel repellent molecules.

scholarly journals On multiplicative degree based topological indices for planar octahedron networks

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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