Multiplicative Zagreb Indices of Molecular Graphs

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.

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Some Algebraic Polynomials and Topological Indices of Octagonal Network

10.20944/preprints201611.0118.v1 ◽

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.

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First reformulated Zagreb indices of some classes of graphs

Carpathian Mathematical Publications ◽

10.15330/cmp.9.2.134-144 ◽

2018 ◽

Vol 9 (2) ◽

pp. 134-144 ◽

Cited By ~ 1

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.

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Computing Analysis of Zagreb Indices for Generalized Sum Graphs under Strong Product

Journal of Chemistry ◽

10.1155/2021/6663624 ◽

2021 ◽

Vol 2021 ◽

pp. 1-20

Author(s):

Muhammad Javaid ◽

Saira Javed ◽

Abdulaziz Mohammed Alanazi ◽

Majdah R. Alotaibi

Keyword(s):

Chemical Engineering ◽

Chemical Properties ◽

Chemical Compounds ◽

Vital Role ◽

Molecular Structures ◽

Zagreb Indices ◽

Graph Theoretic ◽

Strong Product ◽

Physicochemical And Structural Properties ◽

Product Of Graphs

Numerous studies based on mathematical models and tools indicate that there is a strong inherent relationship between the chemical properties of the chemical compounds and drugs with their molecular structures. In the last two decades, the graph-theoretic techniques are frequently used to analyse the various physicochemical and structural properties of the molecular graphs which play a vital role in chemical engineering and pharmaceutical industry. In this paper, we compute Zagreb indices of the generalized sum graphs in the form of the different indices of their factor graphs, where generalized sum graphs are obtained under the operations of subdivision and strong product of graphs. Moreover, the obtained results are illustrated with the help of particular classes of graphs and analysed to find the efficient subclass with dominant indices.

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Some topological indices of dendrimers

International Journal of Computational Materials Science and Engineering ◽

10.1142/s2047684120500189 ◽

2020 ◽

pp. 2050018

Author(s):

S. Alyar ◽

R. Khoeilar ◽

A. Jahanbani

Keyword(s):

Graph Theory ◽

Molecular Graph ◽

Topological Indices ◽

Molecular Structures ◽

Topological Properties ◽

Zagreb Index ◽

Zinc Porphyrin ◽

Molecular Graphs

There are immense applications of graph theory in chemistry and in the study of molecular structures, and after that, it has been increasing exponentially. Molecular graphs have points (vertices) representing atoms and lines (edges) that represent bonds between atoms. In this paper, we study the molecular graph of porphyrin, propyl ether imine, zinc–porphyrin and poly dendrimers and analyzed its topological properties. For this purpose, we have computed topological indices, namely the Albertson index, the sigma index, the Nano-Zagreb index, the first and second hyper [Formula: see text]-indices of porphyrin, propyl ether imine, zinc–porphyrin and poly dendrimers.

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Editorial: Topological investigations of chemical networks

Main Group Metal Chemistry ◽

10.1515/mgmc-2021-0030 ◽

2021 ◽

Vol 44 (1) ◽

pp. 267-269

Author(s):

Muhammad Javaid ◽

Muhammad Imran

Keyword(s):

Information Science ◽

Biological Activities ◽

Chemical Properties ◽

Vital Role ◽

Topological Indices ◽

Molecular Structures ◽

Quantitative Structure ◽

Structure Property ◽

Chemical Structures ◽

Interdisciplinary Field

Abstract The topic of computing the topological indices (TIs) being a graph-theoretic modeling of the networks or discrete structures has become an important area of research nowadays because of its immense applications in various branches of the applied sciences. TIs have played a vital role in mathematical chemistry since the pioneering work of famous chemist Harry Wiener in 1947. However, in recent years, their capability and popularity has increased significantly because of the findings of the different physical and chemical investigations in the various chemical networks and the structures arising from the drug designs. In additions, TIs are also frequently used to study the quantitative structure property relationships (QSPRs) and quantitative structure activity relationships (QSARs) models which correlate the chemical structures with their physio-chemical properties and biological activities in a dataset of chemicals. These models are very important and useful for the research community working in the wider area of cheminformatics which is an interdisciplinary field combining mathematics, chemistry, and information science. The aim of this editorial is to arrange new methods, techniques, models, and algorithms to study the various theoretical and computational aspects of the different types of these topological indices for the various molecular structures.

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F-Index of some graph operations

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500257 ◽

2016 ◽

Vol 08 (02) ◽

pp. 1650025 ◽

Cited By ~ 18

Author(s):

Nilanjan De ◽

Sk. Md. Abu Nayeem ◽

Anita Pal

Keyword(s):

Electron Energy ◽

Topological Index ◽

Zagreb Index ◽

Zagreb Indices ◽

Molecular Graphs ◽

Basic Properties ◽

Graph Operations ◽

Physico Chemical ◽

Vertex Degrees

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total [Formula: see text]-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [A forgotten topological index,J. Math. Chem. 53(4) (2015) 1184–1190.] reinvestigated the index and named it “forgotten topological index” or “F-index”. In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nanostructures.

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Analysis of SC5C7p,q and NPHXp,q Nanotubes via Topological Indices

Journal of Nanomaterials ◽

10.1155/2019/2072789 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Ye-Jun Ge ◽

Jia-Bao Liu ◽

Muhammad Younas ◽

Muhammad Yousaf ◽

Waqas Nazeer

Keyword(s):

Carbon Nanotube ◽

Closed Form ◽

Topological Indices ◽

Molecular Structures ◽

Quantitative Structure ◽

Molecular Graphs ◽

Boiling Points ◽

Structure Activity ◽

Electrical Voltage

Scientists are creating materials, for example, a carbon nanotube-based composite created by NASA that bends when a voltage is connected. Applications incorporate the use of an electrical voltage to change the shape (transform) of air ship wings and different structures. Topological indices are numbers related with molecular graphs to allow quantitative structure activity/property/poisonous relationships. Topological indices catch symmetry of molecular structures and give it a scientific dialect to foresee properties, for example, boiling points, viscosity, and the radius of gyrations. We compute M-polynomials of two nanotubes, SC5C7p,q and NPHXp,q. The closed form of M-polynomials for these nanotubes produces formulas of numerous degree-based topological indices which are functions relying on parameters of the structure and, in combination, decide properties of the concerned nanotubes. Moreover, we sketch our results by using Maple 2015 to see the dependence of our results on the involved parameters.

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

Symmetry ◽

10.3390/sym12061026 ◽

2020 ◽

Vol 12 (6) ◽

pp. 1026 ◽

Cited By ~ 3

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.

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Computational Analysis of new Degree-based descriptors of oxide networks

Open Chemistry ◽

10.1515/chem-2019-0023 ◽

2019 ◽

Vol 17 (1) ◽

pp. 177-182 ◽

Cited By ~ 1

Author(s):

Zafar Hussain ◽

Mobeen Munir ◽

Muhammad Bilal ◽

Alam Ameer ◽

Shazia Rafique ◽

...

Keyword(s):

Computational Analysis ◽

Graphical Analysis ◽

Topological Indices ◽

Zagreb Index ◽

Zagreb Indices ◽

Molecular Graphs ◽

Pharmaceutical Industries ◽

Forgotten Index

AbstractOxide networks have diverse applications in the polymer and pharmaceutical industries. Polynomials and degree-based topological indices have tendencies to correlate properties of molecular graphs. In this article, we formulate the closed forms of Zagreb and forgotten polynomials and topological indices such as Hyper-Zagreb index, first and second multiple Zagreb indices, forgotten index, Albert index, Bell index, IRM(G) of oxide networks. We also compute the F-index of complement of oxide networks, F-coindex of G and F-coindex of complement of oxide networks. We put graphical analysis of each index with respect to the parameter involved in each case.

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Leap Zagreb Connection Numbers for Some Networks Models

Indonesian Journal of Chemistry ◽

10.22146/ijc.53393 ◽

2020 ◽

Vol 20 (6) ◽

pp. 1407 ◽

Cited By ~ 2

Author(s):

Zahid Raza

Keyword(s):

Information Structure ◽

Vital Role ◽

Topological Indices ◽

Chemical Information ◽

Octahedral Structure ◽

First Zagreb Index ◽

Zagreb Index ◽

Aztec Diamond ◽

The First Zagreb Index ◽

Structure Properties

The main object of this study is to determine the exact values of the topological indices which play a vital role in studying chemical information, structure properties like QSAR and QSPR. The first Zagreb index and second Zagreb index are among the most studied topological indices. We now consider analogous graph invariants, based on the second degrees of vertices, called leap Zagreb indices. We compute these indices for Tickysim SpiNNaker model, cyclic octahedral structure, Aztec diamond and extended Aztec diamond.

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