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12 pages, 312 KiB

Open AccessArticle

Comparative Study of Planar Octahedron Molecular Structure via Eccentric Invariants

by Zheng-Qing Chu, Haidar Ali, Didar Abdulkhaleq Ali, Muhammad Nadeem, Syed Ajaz K. Kirmani and Parvez Ali

Molecules 2023, 28(2), 556; https://doi.org/10.3390/molecules28020556 - 5 Jan 2023

Cited by 1 | Viewed by 2004

Abstract

A branch of graph theory that makes use of a molecular graph is called chemical graph theory. Chemical graph theory is used to depict a chemical molecule. A graph is connected if there is an edge between every pair of vertices. A topological [...] Read more.

A branch of graph theory that makes use of a molecular graph is called chemical graph theory. Chemical graph theory is used to depict a chemical molecule. A graph is connected if there is an edge between every pair of vertices. A topological index is a numerical value related to the chemical structure that claims to show a relationship between chemical structure and various physicochemical attributes, chemical reactivity, or, you could say, biological activity. In this article, we examined the topological properties of a planar octahedron network of m dimensions and computed the total eccentricity, average eccentricity, Zagreb eccentricity, geometric arithmetic eccentricity, and atom bond connectivity eccentricity indices, which are used to determine the distance between the vertices of a planar octahedron network. Full article

(This article belongs to the Special Issue Study of Molecules in the Light of Spectral Graph Theory)

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17 pages, 1718 KiB

Open AccessArticle

Measurement of Composites Index on Low Carbon Development Supporting Food Security

by Dwi Sartika Adetama, Akhmad Fauzi, Bambang Juanda and Dedi Budiman Hakim

Sustainability 2021, 13(23), 13352; https://doi.org/10.3390/su132313352 - 2 Dec 2021

Cited by 2 | Viewed by 2174

Abstract

Since the concept of low carbon development (LCD) was adopted at the 1992 Rio de Janeiro Earth Summit, Indonesia has been committed to implementing the reduction of greenhouse gas emissions. In 2020, the country issued Presidential Regulation No. 18, which made LCD one [...] Read more.

Since the concept of low carbon development (LCD) was adopted at the 1992 Rio de Janeiro Earth Summit, Indonesia has been committed to implementing the reduction of greenhouse gas emissions. In 2020, the country issued Presidential Regulation No. 18, which made LCD one of the national priority programs to maintain economic and social through low emission activities and reduce the overexploitation of natural resources. The LCD is a way for the country to overcome the tradeoff between economic growth and environmental degradation. Nevertheless, LCD is a new initiative for Indonesia, so it needs strategic indicators that influence the achievement of development. This paper attempts to integrate macro-regional development indicators that combine each region’s gross domestic product, human development index, and unemployment rate with LCD indicators, including the environmental quality index and g reenhouse gas emissions. The combined indicators were constructed by composite index through the Shannon entropy method, geometric and arithmetic means using the technique for order preference by similarity to ideal solution (TOPSIS) method. The results show significant differences among provinces concerning to macro-regional indicators once the LCD indicators were incorporated. The results of this analysis could be used by policymakers to evaluate the green development of regions. Full article

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17 pages, 334 KiB

Open AccessArticle

Some Properties of the Arithmetic–Geometric Index

by Edil D. Molina, José M. Rodríguez, José L. Sánchez and José M. Sigarreta

Symmetry 2021, 13(5), 857; https://doi.org/10.3390/sym13050857 - 12 May 2021

Cited by 15 | Viewed by 2374

Abstract

Recently, the arithmetic–geometric index (AG) was introduced, inspired by the well-known and studied geometric–arithmetic index (GA). In this work, we obtain new bounds on the arithmetic–geometric index, improving upon some already known bounds. In particular, we show families of graphs where such bounds [...] Read more.

Recently, the arithmetic–geometric index (AG) was introduced, inspired by the well-known and studied geometric–arithmetic index (GA). In this work, we obtain new bounds on the arithmetic–geometric index, improving upon some already known bounds. In particular, we show families of graphs where such bounds are attained. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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22 pages, 749 KiB

Open AccessArticle

Computing Degree Based Topological Properties of Third Type of Hex-Derived Networks

by Chang-Cheng Wei, Haidar Ali, Muhammad Ahsan Binyamin, Muhammad Nawaz Naeem and Jia-Bao Liu

Mathematics 2019, 7(4), 368; https://doi.org/10.3390/math7040368 - 23 Apr 2019

Cited by 26 | Viewed by 3811

Abstract

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical [...] Read more.

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical network. Simonraj et al. derived a new type of graphs, which is named a third type of hex-derived networks. In our work, we discuss the third type of hex-derived networks

H D N 3 (r)

,

T H D N 3 (r)

,

R H D N 3 (r)

,

C H D N 3 (r)

, and compute exact results for topological indices which are based on degrees of end vertices. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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16 pages, 346 KiB

Open AccessArticle

Topological Indices of mth Chain Silicate Graphs

by Jia-Bao Liu, Muhammad Kashif Shafiq, Haidar Ali, Asim Naseem, Nayab Maryam and Syed Sheraz Asghar

Mathematics 2019, 7(1), 42; https://doi.org/10.3390/math7010042 - 4 Jan 2019

Cited by 13 | Viewed by 3603

Abstract

A topological index is a numerical representation of a chemical structure, while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its numerical representation. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, and biological activity are [...] Read more.

A topological index is a numerical representation of a chemical structure, while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its numerical representation. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, and biological activity are determined by the chemical applications of graph theory. The biological activity of chemical compounds can be constructed by the help of topological indices such as atom-bond connectivity (ABC), Randić, and geometric arithmetic (GA). In this paper, Randić, atom bond connectivity (ABC), Zagreb, geometric arithmetic (GA), ABC₄, and GA₅ indices of the mth chain silicate

S L (m, n)

network are determined. Full article

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18 pages, 2058 KiB

Open AccessArticle

On Degree-Based Topological Indices of Symmetric Chemical Structures

by Jia-Bao Liu, Haidar Ali, Muhammad Kashif Shafiq and Usman Munir

Symmetry 2018, 10(11), 619; https://doi.org/10.3390/sym10110619 - 9 Nov 2018

Cited by 9 | Viewed by 3805

Abstract

A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph [...] Read more.

A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study HDCN1(m,n) and HDCN2(m,n) of dimension

m, n

and derive analytical closed results of general Randić index

R_{α} (G)

for different values of

α

. We also compute the general first Zagreb, ABC, GA,

A B C_{4}

and

G A_{5}

indices for these Hex derived cage networks for the first time and give closed formulas of these degree-based indices. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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15 pages, 335 KiB

Open AccessArticle

Computation of Topological Indices of Some Special Graphs

by Mohammed Salaheldeen Abdelgader, Chunxiang Wang and Sarra Abdalrhman Mohammed

Mathematics 2018, 6(3), 33; https://doi.org/10.3390/math6030033 - 1 Mar 2018

Cited by 19 | Viewed by 6572

Abstract

There are several chemical indices that have been introduced in theoretical chemistry to measure the properties of molecular topology, such as distance-based topological indices, degree-based topological indices and counting-related topological indices. Among the degree-based topological indices, the atom-bond connectivity ( [...] Read more.

There are several chemical indices that have been introduced in theoretical chemistry to measure the properties of molecular topology, such as distance-based topological indices, degree-based topological indices and counting-related topological indices. Among the degree-based topological indices, the atom-bond connectivity (

A B C

) index and geometric–arithmetic (

G A

) index are the most important, because of their chemical significance. Certain physicochemical properties, such as the boiling point, stability and strain energy, of chemical compounds are correlated by these topological indices. In this paper, we study the molecular topological properties of some special graphs. The indices

(A B C)

,

(A B C_{4})

,

(G A)

and

(G A_{5})

of these special graphs are computed. Full article

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10 pages, 903 KiB

Open AccessArticle

On Topological Indices of Certain Families of Nanostar Dendrimers

by Mohamad Nazri Husin, Roslan Hasni, Nabeel Ezzulddin Arif and Muhammad Imran

Molecules 2016, 21(7), 821; https://doi.org/10.3390/molecules21070821 - 24 Jun 2016

Cited by 43 | Viewed by 7044

Abstract

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 [...] Read more.

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 Randić connectivity index, atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are used to predict the bioactivity of different chemical compounds. A dendrimer is an artificially manufactured or synthesized molecule built up from the branched units called monomers. In this paper, the fourth version of ABC index and the fifth version of GA index of certain families of nanostar dendrimers are investigated. We derive the analytical closed formulas for these families of nanostar dendrimers. The obtained results can be of use in molecular data mining, particularly in researching the uniqueness of tested (hyper-branched) molecular graphs. Full article

(This article belongs to the Special Issue Functional Dendrimers)

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