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24 pages, 433 KB

Open AccessArticle

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

by Juan L. G. Guirao, Muhammad Imran, Muhammad Kamran Siddiqui and Shehnaz Akhter

Symmetry 2020, 12(6), 1026; https://doi.org/10.3390/sym12061026 - 17 Jun 2020

Cited by 16 | Viewed by 2810

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices)

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11 pages, 2026 KB

Open AccessArticle

Topological Characterization of the Crystallographic Structure of Titanium Difluoride and Copper (I) Oxide

by Hong Yang, Mehwish Hussain Muhammad, Muhammad Aamer Rashid, Sarfraz Ahmad, Muhammad Kamran Siddiqui and Muhammad Naeem

Atoms 2019, 7(4), 100; https://doi.org/10.3390/atoms7040100 - 1 Nov 2019

Cited by 4 | Viewed by 2807

Abstract

Owing to their distinguished properties, titanium difluoride (TiF₂) and the crystallographic structure of Cu₂O have attracted a great deal of attention in the field of quantitative structure–property relationships (QSPRs) in recent years. A topological index of a diagram (G) [...] Read more.

Owing to their distinguished properties, titanium difluoride (TiF₂) and the crystallographic structure of Cu₂O have attracted a great deal of attention in the field of quantitative structure–property relationships (QSPRs) in recent years. A topological index of a diagram (G) is a numerical quantity identified with G which portrays the sub-atomic chart G. In 1972, Gutman and Trinajstić resented the first and second Zagreb topological files of atomic diagrams. In this paper, we determine a hyper-Zagreb list, a first multiple Zagreb file, a second different Zagreb record, and Zagreb polynomials for titanium difluoride (TiF₂) and the crystallographic structure of Cu₂O. Full article

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10 pages, 607 KB

Open AccessArticle

Maximizing and Minimizing Multiplicative Zagreb Indices of Graphs Subject to Given Number of Cut Edges

by Shaohui Wang, Chunxiang Wang, Lin Chen, Jia-Bao Liu and Zehui Shao

Mathematics 2018, 6(11), 227; https://doi.org/10.3390/math6110227 - 29 Oct 2018

Cited by 8 | Viewed by 3048

Abstract

Given a (molecular) graph, the first multiplicative Zagreb index

Π_{1}

is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index

Π_{2}

is expressed as the product of endvertex degree of each [...] Read more.

Given a (molecular) graph, the first multiplicative Zagreb index

Π_{1}

is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index

Π_{2}

is expressed as the product of endvertex degree of each edge over all edges. We consider a set of graphs

G_{n, k}

having n vertices and k cut edges, and explore the graphs subject to a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs in

G_{n, k}

are provided. We also provide these graphs with the largest and smallest

Π_{1} (G)

and

Π_{2} (G)

in

G_{n, k}

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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11 pages, 487 KB

Open AccessArticle

Topological Characterizations and Index-Analysis of New Degree-Based Descriptors of Honeycomb Networks

by Zafar Hussain, Mobeen Munir, Shazia Rafique and Shin Min Kang

Symmetry 2018, 10(10), 478; https://doi.org/10.3390/sym10100478 - 11 Oct 2018

Cited by 14 | Viewed by 3041

Abstract

Topological indices and connectivity polynomials are invariants of molecular graphs. These invariants have the tendency of predicting the properties of the molecular structures. The honeycomb network structure is an important type of benzene network. In the present article, new topological characterizations of honeycomb [...] Read more.

Topological indices and connectivity polynomials are invariants of molecular graphs. These invariants have the tendency of predicting the properties of the molecular structures. The honeycomb network structure is an important type of benzene network. In the present article, new topological characterizations of honeycomb networks are given in the form of degree-based descriptors. In particular, we compute Zagreb and Forgotten polynomials and some topological indices such as the hyper-Zagreb index, first and second multiple Zagreb indices and the Forgotten index, F. We, for the first time, determine some regularity indices such as the Albert index, Bell index and

I R M (G)

index, as well as the F-index of the complement of the honeycomb network and several co-indices related to this network without considering the graph of its complement or even the line graph. These indices are useful for correlating the physio-chemical properties of the honeycomb network. We also give a graph theoretic analysis of some indices against the dimension of this network. Full article

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10 pages, 4227 KB

Open AccessArticle

Computing Topological Indices and Polynomials for Line Graphs

by Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran and Muhammad Faisal Nadeem

Mathematics 2018, 6(8), 137; https://doi.org/10.3390/math6080137 - 10 Aug 2018

Cited by 12 | Viewed by 3971

Abstract

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the [...] Read more.

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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11 pages, 3129 KB

Open AccessArticle

On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si₂C₃-I[p,q] and Si₂C₃-II[p,q]

by Young Chel Kwun, Abaid Ur Rehman Virk, Waqas Nazeer, M. A. Rehman and Shin Min Kang

Symmetry 2018, 10(8), 320; https://doi.org/10.3390/sym10080320 - 3 Aug 2018

Cited by 36 | Viewed by 5366

Abstract

The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topological indices help us to predict many physico-chemical [...] Read more.

The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topological indices help us to predict many physico-chemical properties of the concerned molecular compound. In this article, we compute Generalized first and multiplicative Zagreb indices, the multiplicative version of the atomic bond connectivity index, and the Generalized multiplicative Geometric Arithmetic index for silicon-carbon

S i_{2} C_{3^{-}} I [p, q]

and

S i_{2} C_{3^{-}} II [p, q]

second. Full article

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16 pages, 25520 KB

Open AccessArticle

Computing Zagreb Indices and Zagreb Polynomials for Symmetrical Nanotubes

by Zehui Shao, Muhammad Kamran Siddiqui and Mehwish Hussain Muhammad

Symmetry 2018, 10(7), 244; https://doi.org/10.3390/sym10070244 - 28 Jun 2018

Cited by 129 | Viewed by 4933

Abstract

Topological indices are numbers related to sub-atomic graphs to allow quantitative structure-movement/property/danger connections. These topological indices correspond to some specific physico-concoction properties such as breaking point, security, strain vitality of chemical compounds. The idea of topological indices were set up in compound graph [...] Read more.

Topological indices are numbers related to sub-atomic graphs to allow quantitative structure-movement/property/danger connections. These topological indices correspond to some specific physico-concoction properties such as breaking point, security, strain vitality of chemical compounds. The idea of topological indices were set up in compound graph hypothesis in view of vertex degrees. These indices are valuable in the investigation of mitigating exercises of specific Nanotubes and compound systems. In this paper, we discuss Zagreb types of indices and Zagreb polynomials for a few Nanotubes covered by cycles. Full article

(This article belongs to the Special Issue Symmetry and Complexity)

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12 pages, 1458 KB

Open AccessArticle

Topological Characterization of Carbon Graphite and Crystal Cubic Carbon Structures

by Wei Gao, Muhammad Kamran Siddiqui, Muhammad Naeem and Najma Abdul Rehman

Molecules 2017, 22(9), 1496; https://doi.org/10.3390/molecules22091496 - 7 Sep 2017

Cited by 133 | Viewed by 5414

Abstract

Graph theory is used for modeling, designing, analysis and understanding chemical structures or chemical networks and their properties. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this article, we study the [...] Read more.

Graph theory is used for modeling, designing, analysis and understanding chemical structures or chemical networks and their properties. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this article, we study the chemical graphs of carbon graphite and crystal structure of cubic carbon. Moreover, we compute and give closed formulas of degree based additive topological indices, namely hyper-Zagreb index, first multiple and second multiple Zagreb indices, and first and second Zagreb polynomials. Full article

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15 pages, 7017 KB

Open AccessArticle

Some Invariants of Jahangir Graphs

by Mobeen Munir, Waqas Nazeer, Shin Min Kang, Muhammad Imran Qureshi, Abdul Rauf Nizami and Youl Chel Kwun

Symmetry 2017, 9(1), 17; https://doi.org/10.3390/sym9010017 - 23 Jan 2017

Cited by 22 | Viewed by 6964

Abstract

In this report, we compute closed forms of M-polynomial, first and second Zagreb polynomials and forgotten polynomial for Jahangir graphs J_n,_m for all values of m and n. From the M-polynomial, we recover many degree-based topological indices such as [...] Read more.

In this report, we compute closed forms of M-polynomial, first and second Zagreb polynomials and forgotten polynomial for Jahangir graphs J_n,_m for all values of m and n. From the M-polynomial, we recover many degree-based topological indices such as first and second Zagreb indices, modified Zagreb index, Symmetric division index, etc. We also compute harmonic index, first and second multiple Zagreb indices and forgotten index of Jahangir graphs. Our results are extensions of many existing results. Full article

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Graphical abstract

12 pages, 4501 KB

Open AccessArticle

Some Algebraic Polynomials and Topological Indices of Generalized Prism and Toroidal Polyhex Networks

by Muhammad Ajmal, Waqas Nazeer, Mobeen Munir, Shin Min Kang and Young Chel Kwun

Symmetry 2017, 9(1), 5; https://doi.org/10.3390/sym9010005 - 29 Dec 2016

Cited by 25 | Viewed by 5459

Abstract

A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this report, we compute closed forms of first [...] Read more.

A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this report, we compute closed forms of first Zagreb, second Zagreb, and forgotten polynomials of generalized prism and toroidal polyhex networks. We also compute hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and forgotten index of these networks. Moreover we gave graphical representation of our results, showing the technical dependence of each topological index and polynomial on the involved structural parameters. Full article

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