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 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.

scholarly journals The Generalized Zagreb Index of Some Carbon Structures

2018 ◽  
Vol 26 (1) ◽  
pp. 91-104 ◽  
Author(s):  
Prosanta Sarkar ◽  
Nilanjan De ◽  
Anita Pal
Keyword(s):  
Graph Theory ◽  
Physicochemical Properties ◽  
Topological Indices ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Chemical Structures ◽  
Chemical Graph Theory ◽  
Carbon Structures

Abstract In chemical graph theory, chemical structures are model edthrough a graph where atoms are considered as vertices and edges are bonds between them. In chemical sciences, topological indices are used for understanding the physicochemical properties of molecules. In this work, we study the generalized Zagreb index for three types of carbon allotrope’s theoretically.

scholarly journals On Computation Degree-Based Topological Descriptors for Planar Octahedron Networks

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.

On Certain Topological Indices of Three-Layered Single-Walled Titania Nanosheets

2020 ◽  
Vol 23 ◽  
Author(s):  
Micheal Arockiaraj ◽  
Jia-Bao Liu ◽  
M. Arulperumjothi ◽  
S. Prabhu
Keyword(s):  
Materials Science ◽  
Molecular Graph ◽  
Quantitative Structure Activity Relationship ◽  
Topological Indices ◽  
Quantitative Structure ◽  
Structure Property ◽  
Mathematical Properties ◽  
Chemical Graph Theory ◽  
Research Article ◽  
Cut Method

Aim and Objective: Nanostructures are objects whose sizes are between microscopic and molecular. The most significant of these new elements are carbon nanotubes. These elements have extraordinary microelectronic properties and many other exclusive physiognomies. Recently, researchers have given the attention to the mathematical properties of these materials. The aim and objective of this research article is to investigate the most important molecular descriptors namely Wiener, edge-Wiener, vertex-edge-Wiener, vertex-Szeged, edge-Szeged, edge-vertex-Szeged, total-Szeged, PI, Schultz, Gutman, Mostar, edge-Mostar, and total-Mostar indices of three-layered single-walled titania nanosheets. By computing these topological indices, materials science researchers can have a better understanding of structural and physical properties of titania nanosheets, and thereby more easily synthesizing new variants of titania nanosheets with more amenable physicochemical properties. Methods: The cut method turned out to be extremely handy when dealing with distance-based graph invariants which are in turn among the central concepts of chemical graph theory. In this method, we use the Djokovic ́-Winkler relation to find the suitable edge cuts to leave the graph into exactly two components. Based on the graph theoretical measures of the components, we obtain the desired topological indices by mathematical computations. Results: In this paper, distance-based indices for three-layered single-walled titania nanosheets were investigated and given the exact expressions for various dimensions of three-layered single-walled titania nanosheets. These indices may be useful in synthesizing new variants of titania nanosheets and the computed topological indices play an important role in studies of Quantitative structure-activity relationship (QSAR) and Quantitative structure-property relationship (QSPR). Conclusion: In this paper, we have obtained the closed expressions of several distance-based topological indices of three-layered single-walled titania nanosheet TNS_3 [m,n] molecular graph for the cases m≥ n and m < n. The graphical validations for the computed indices are done and we observe that the Wiener types, Schultz and Gutman indices perform in a similar way whereas PI and Mostar type indices perform in the same way.

scholarly journals Molecular description of copper(II) oxide

2017 ◽  
Vol 36 (1) ◽  
Author(s):  
Mohammad Reza Farahani ◽  
Wei Gao ◽  
Abdul Qudair Baig ◽  
Wasaq Khalid
Keyword(s):  
Graph Theory ◽  
Molecular Graph ◽  
Theoretical Chemistry ◽  
Topological Descriptors ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Mathematical Chemistry ◽  
Chemical Graph Theory ◽  
Numerical Invariants ◽  
Molecular Description

Graph theory has much advancement in the field of mathematical chemistry. Recently, chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields.In this article, we study the chemical graph of copper oxide and compute degree based topological indices mainly ABC, GA, ABC4, GA5, general Randić index and Zagreb index for copper(II) oxide, CuO. Furthermore, we give exact formulas of these indices which are helpful in studying the underlying topologies.

scholarly journals On the maximum and minimum first reformulated Zagreb index of graphs with connectivity at most k

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 75-83 ◽  
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Lan Xu ◽  
Beibei Ma
Keyword(s):  
Graph Theory ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Zagreb Indices ◽  
Chemical Graph Theory

The authors Milicevic et al. introduced the reformulated Zagreb indices [1], which is a generalization of classical Zagreb indices of chemical graph theory. In this paper, we mainly consider the maximum and minimum for the first reformulated index of graphs with connectivity at most k. The corresponding extremal graphs are characterized.

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.

Tetracyclic graphs with maximum second Zagreb index: A simple approach

2018 ◽  
Vol 11 (05) ◽  
pp. 1850064 ◽  
Author(s):  
Akbar Ali
Keyword(s):  
Graph Theory ◽  
Short Note ◽  
Topological Indices ◽  
Simple Approach ◽  
Graph Invariants ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Chemical Graph Theory

In the chemical graph theory, graph invariants are usually referred to as topological indices. The second Zagreb index (denoted by [Formula: see text]) is one of the most studied topological indices. For [Formula: see text], let [Formula: see text] be the collection of all non-isomorphic connected graphs with [Formula: see text] vertices and [Formula: see text] edges (such graphs are known as tetracyclic graphs). Recently, Habibi et al. [Extremal tetracyclic graphs with respect to the first and second Zagreb indices, Trans. on Combin. 5(4) (2016) 35–55.] characterized the graph having maximum [Formula: see text] value among all members of the collection [Formula: see text]. In this short note, an alternative but relatively simple approach is used for characterizing the aforementioned graph.

scholarly journals Review on Chemical Graph Theory and Its Application in Computer-Assisted Structure Elucidation

2021 ◽  
Author(s):  
Mehmet Aziz Yirik ◽  
Kumsal Ecem Colpan ◽  
Saskia Schmidt ◽  
Maria Sorokina ◽  
Christoph Steinbeck
Keyword(s):  
Graph Theory ◽  
Data Structures ◽  
Computer Assisted ◽  
Structure Property ◽  
Chemical Graph ◽  
Chemical Structures ◽  
Chemical Graph Theory ◽  
Structure Activity ◽  
History Of ◽  
Computer Assisted Structure Elucidation

The chemical graph theory is a subfield of mathematical chemistry which applies classic graph theory to chemical entities and phenomena. Chemical graphs are main data structures to represent chemical structures in cheminformatics. Computable properties of graphs lay the foundation for (quantitative) structure activity and structure property predictions - a core discipline of cheminformatics. It has a historic relevance for natural sciences, such as chemistry, biochemistry and biology, and is in the heart of modern disciplines, such as cheminformatics and bioinformatics. This review first covers the history of chemical graph theory, then provides an overview of its various techniques and applications for CASE, and finally summarises modern tools using chemical graph theory for CASE.

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.

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