1
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Hussain M, Siddiqui MK, Hanif MF, Mahmood H, Saddique Z, Asefa Fufa S. On K-Banhatti indices and entropy measure for rhodium (III) chloride via linear regression models. Heliyon 2023; 9:e20935. [PMID: 37916098 PMCID: PMC10616341 DOI: 10.1016/j.heliyon.2023.e20935] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2023] [Revised: 09/27/2023] [Accepted: 10/11/2023] [Indexed: 11/03/2023] Open
Abstract
Rhodium (III) chloride is a metallic compound characterized by its shiny and silvery-white appearance. It possesses high reflectivity and exhibits excellent resistance to corrosion. This makes it a popular choice for applications such as plating materials in jewelry and other decorative items, imparting a lustrous and reflective surface to the coated objects. Topological indices are numerical parameters employed to characterize the topology of a molecular structure. These indices are derived from the connectivity of atoms within the molecule and serve as predictors for various molecular properties, including reactivity, stability, and solubility. On the other hand, the Shannon entropy of a graph finds extensive applications in network science. It is utilized in the analysis of diverse networks, such as social networks, biological networks, and transportation networks. The Shannon entropy allows for the characterization of a network's topology and structure, aiding in the identification of crucial nodes or structures that play significant roles in network functionality and stability. In this paper, our primary objective is to compute different K-Banhatti indices and employ them to evaluate the entropy measure of Rhodium (III) chloride R h C l 3 . Additionally, we conducted an examination through linear regression analysis involving various indices and entropies associated with Rhodium chloride. Moreover, we established a correlation between degree-based Banhatti indices and entropies via the line fit method.
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Affiliation(s)
- Mazhar Hussain
- Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
| | | | - Muhammad Farhan Hanif
- Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan
| | - Hasan Mahmood
- Department of Mathematics, Government College University, Lahore, Pakistan
| | - Zohaib Saddique
- Department of Chemistry, University of the Punjab, Quaid-i-Azam Campus Lahore, Pakistan
| | - Samuel Asefa Fufa
- Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia
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2
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Tang Y, Labba M, Jamil MK, Azeem M, Zhang X. Edge valency-based entropies of tetrahedral sheets of clay minerals. PLoS One 2023; 18:e0288931. [PMID: 37478115 PMCID: PMC10361463 DOI: 10.1371/journal.pone.0288931] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2023] [Accepted: 07/06/2023] [Indexed: 07/23/2023] Open
Abstract
Humanity has always benefited from an intercapillary study in the quantification of natural occurrences in mathematics and other pure scientific fields. Graph theory was extremely helpful to other studies, particularly in the applied sciences. Specifically, in chemistry, graph theory made a significant contribution. For this, a transformation is required to create a graph representing a chemical network or structure, where the vertices of the graph represent the atoms in the chemical compound and the edges represent the bonds between the atoms. The quantity of edges that are incident to a vertex determines its valency (or degree) in a graph. The degree of uncertainty in a system is measured by the entropy of a probability. This idea is heavily grounded in statistical reasoning. It is primarily utilized for graphs that correspond to chemical structures. The development of some novel edge-weighted based entropies that correspond to valency-based topological indices is made possible by this research. Then these compositions are applied to clay mineral tetrahedral sheets. Since they have been in use for so long, corresponding indices are thought to be the most effective methods for quantifying chemical graphs. This article develops multiple edge degree-based entropies that correlate to the indices and determines how to modify them in order to assess the significance of each type.
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Affiliation(s)
- Yong Tang
- School of Computer Science, Chengdu University, Chengdu, China
| | - Muhammad Labba
- Department of Mathematics, Riphah International University Lahore, Lahore, Pakistan
| | | | - Muhammad Azeem
- School of Computer Science, Chengdu University, Chengdu, China
| | - Xiujun Zhang
- School of Computer Science, Chengdu University, Chengdu, China
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3
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Arockiaraj M, Paul D, Ghani MU, Tigga S, Chu YM. Entropy structural characterization of zeolites BCT and DFT with bond-wise scaled comparison. Sci Rep 2023; 13:10874. [PMID: 37407626 DOI: 10.1038/s41598-023-37931-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2023] [Accepted: 06/29/2023] [Indexed: 07/07/2023] Open
Abstract
Entropy of a connected network is a quantitative measure from information theory that has triggered a plethora of research domains in molecular chemistry, biological sciences and computer programming due to its inherent capacity to explore the structural characteristics of complex molecular frameworks that have low structural symmetry as well as high diversity. The analysis of the structural order is greatly simplified through the topological indices based graph entropy metrics, which are then utilized to predict the structural features of molecular frameworks. This predictability has not only revolutionized the study of zeolitic frameworks but has also given rise to new generations of frameworks. We make a comparative study of two versatile framework topologies namely zeolites BCT and DFT, which have been widely utilized to create a new generation of frameworks known as metal organic frameworks. We discuss bond-additive topological indices and compute entropy measure descriptors for zeolites BCT and DFT using degree and degree-sum parameters. In addition, we perform bond-wise scaled comparative analysis between BCT and DFT which shows that zeolite BCT has greater entropy values compared to zeolite DFT.
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Affiliation(s)
| | - Daniel Paul
- Department of Mathematics, Sri Sairam Institute of Technology, Chennai, 600044, India
| | - Muhammad Usman Ghani
- Institute of Mathematics, Khawaja Fareed University of Engineering & Information Technology, Abu Dhabi Road, Rahim Yar Khan, 64200, Pakistan
| | - Sushil Tigga
- Department of Mathematics, Loyola College, Chennai, 600034, India
| | - Yu-Ming Chu
- Department of Mathematics, Huzhou University, Huzhou, 313000, People's Republic of China.
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4
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Raza Z, Arockiaraj M, Maaran A, Kavitha SRJ, Balasubramanian K. Topological Entropy Characterization, NMR and ESR Spectral Patterns of Coronene-Based Transition Metal Organic Frameworks. ACS OMEGA 2023; 8:13371-13383. [PMID: 37065084 PMCID: PMC10099125 DOI: 10.1021/acsomega.3c00825] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/08/2023] [Accepted: 03/08/2023] [Indexed: 06/19/2023]
Abstract
Metal organic frameworks (MOFs) are topical crystalline materials with high porosity and inner surface areas synthesized from naturally occurring minerals. Such MOFs with transition metals have attracted considerable attention because of their fascinating morphological diversity and tunable characteristics. The coronene-based structural frameworks with transition metal atoms are synthesized by repeating a fixed coronene unit at several levels. In this study, topological indices and NMR and ESR spectral patterns are computed for these MOFs to shed light on their structures and spectral properties. We obtained mathematical expressions of topological indices based on degree and degree-sum values of MOFs for the rectangular, hexagonal, and parallelogram peripheral shapes. Furthermore, the entropy measures of these novel frameworks are evaluated with the help of index functionals and compared to a wide range of degree-based descriptors. The NMR and ESR spectral patterns have been obtained from the distance degree vector sequences and symmetries for the three MOFs.
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Affiliation(s)
- Zahid Raza
- Department of Mathematics,
College of Sciences, University of Sharjah, Sharjah 27272 United Arab
Emirates
| | | | - Aravindan Maaran
- Department of Mathematics, Loyola College, Chennai 600034, India
| | | | - Krishnan Balasubramanian
- School of Molecular Sciences, Arizona State
University, Tempe, Arizona 85287-1604, United States
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5
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Arockiaraj M, Kavitha SRJ, Klavžar S, Fiona JC, Balasubramanian K. Topological, Spectroscopic and Energetic Properties of Cycloparaphenylene Series. Polycycl Aromat Compd 2023. [DOI: 10.1080/10406638.2023.2186442] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/14/2023]
Affiliation(s)
| | | | - Sandi Klavžar
- Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
- Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia
- Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
| | - J. Celin Fiona
- Department of Mathematics, Loyola College, Chennai, India
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6
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On Neighborhood Degree-Based Topological Analysis over Melamine-Based TriCF Structure. Symmetry (Basel) 2023. [DOI: 10.3390/sym15030635] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/06/2023] Open
Abstract
Triazine-based covalent organic frameworks (TriCFs) were synthesized using melamine, and cyanuric acid is a brand-new synthetic lubricant, which is thermo-stable and possesses a lamellar structure. This article demonstrates how topological descriptors for the TriCF structure are precisely evaluated using the degree sum of the end vertex neighbors and also some molecular descriptors with multiplicative neighborhood degree sums are evaluated. Furthermore, the neighborhood entropy measures for the outcomes are provided. The results are compared using the graph theoretical method.
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7
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Huang Q, Labba M, Azeem M, Jamil MK, Luo R. Tetrahedral sheets of clay minerals and their edge valency-based entropy measures. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:8068-8084. [PMID: 37161186 DOI: 10.3934/mbe.2023350] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
Humanity has always benefited from an intercapillary study in the quantification of natural occurrences in mathematics and other pure scientific fields. Graph theory was extremely helpful to other studies, particularly in the applied sciences. Specifically, in chemistry, graph theory made a significant contribution. For this, a transformation is required to create a graph representing a chemical network or structure, where the vertices of the graph represent the atoms in the chemical compound and the edges represent the bonds between the atoms. The quantity of edges that are incident to a vertex determines its valency (or degree) in a graph. The degree of uncertainty in a system is measured by the entropy of a probability. This idea is heavily grounded in statistical reasoning. It is primarily utilized for graphs that correspond to chemical structures. The development of some novel edge-weighted based entropies that correspond to valency-based topological indices is made possible by this research. Then these compositions are applied to clay mineral tetrahedral sheets. Since they have been in use for so long, corresponding indices are thought to be the most effective methods for quantifying chemical graphs. This article develops multiple edge degree-based entropies that correlate to the indices and determines how to modify them to assess the significance of each type.
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Affiliation(s)
- Qingqun Huang
- School of Mathematics and Physics, Hechi University, Yizhou, Guangxi 456300, China
| | - Muhammad Labba
- Department of Mathematics, Riphah International University Lahore, Pakistan
| | - Muhammad Azeem
- Department of Mathematics, Riphah International University Lahore, Pakistan
| | | | - Ricai Luo
- School of Mathematics and Physics, Hechi University, Yizhou, Guangxi 456300, China
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8
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Augustine T, Roy S, Sahaya Vijay J, Maria Thomas J, Shanmugam P. Topological study on degree based molecular descriptors of fullerene cages. Mol Phys 2023. [DOI: 10.1080/00268976.2023.2179858] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/19/2023]
Affiliation(s)
- Tony Augustine
- Department of Mathematics, Vellore Institute of Technology, Vellore, India
| | - S. Roy
- Department of Mathematics, Vellore Institute of Technology, Vellore, India
| | - J. Sahaya Vijay
- Department of Mathematics, Vellore Institute of Technology, Vellore, India
| | - Jain Maria Thomas
- Department of Chemistry, St. George's College, Aruvithura, Kerela, India
| | - P. Shanmugam
- Department of Chemistry, Thammasat University, Pathumthani, Thailand
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9
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Alqahtani Y, Jamil MK, Alshehri H, Ahmad A, Azeem M. Vertex metric resolvability of COVID antiviral drug structures. JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2023. [DOI: 10.3233/jifs-220964] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023]
Abstract
In November of 2019 year, there was the first case of COVID-19 (Coronavirus) recorded, and up to 3rd of April of 2020, 1,116,643 confirmed positive cases, and around 59,158 dying were recorded. Novel antiviral structures of the 2019 pandemic disease Coronavirus are discussed in terms of the metric basis of their molecular graph. These structures are named arbidol, chloroquine, hydroxy-chloroquine, thalidomide, and theaflavin. Metric dimension or metric basis is a concept in which the whole vertex set of a structure is uniquely identified with a chosen subset named as resolving set. Moreover, the fault-tolerant concept of those structures is also included in this study. By this concept of vertex-metric resolvability of COVID antiviral drug structures are uniquely identified and help to study the structural properties of the structure.
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Affiliation(s)
- Yahya Alqahtani
- College of Computer Science and Information , Jazan University, Jazan, Saudi Arabia
| | - Muhammad Kamran Jamil
- Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University Lahore, Pakistan
| | - Hamdan Alshehri
- College of Computer Science and Information , Jazan University, Jazan, Saudi Arabia
| | - Ali Ahmad
- College of Computer Science and Information , Jazan University, Jazan, Saudi Arabia
| | - Muhammad Azeem
- Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University Lahore, Pakistan
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10
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Huang Q, Khalil A, Ali DA, Ahmad A, Luo R, Azeem M. Breast cancer chemical structures and their partition resolvability. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:3838-3853. [PMID: 36899607 DOI: 10.3934/mbe.2023180] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
Cancer is a disease that causes abnormal cell formation and spreads throughout the body, causing harm to other organs. Breast cancer is the most common kind among many of cancers worldwide. Breast cancer affects women due to hormonal changes or genetic mutations in DNA. Breast cancer is one of the primary causes of cancer worldwide and the second biggest cause of cancer-related deaths in women. Metastasis development is primarily linked to mortality. Therefore, it is crucial for public health that the mechanisms involved in metastasis formation are identified. Pollution and the chemical environment are among the risk factors that are being indicated as impacting the signaling pathways involved in the construction and growth of metastatic tumor cells. Due to the high risk of mortality of breast cancer, breast cancer is potentially fatal, more research is required to tackle the deadliest disease. We considered different drug structures as chemical graphs in this research and computed the partition dimension. This can help to understand the chemical structure of various cancer drugs and develop formulation more efficiently.
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Affiliation(s)
- Qingqun Huang
- School of Mathematics and Physics, Hechi University, Yizhou, Guangxi 456300, China
| | - Adnan Khalil
- Department Computer Sciences, Al-Razi Institute Saeed Park, Lahore Pakistan
| | | | - Ali Ahmad
- College of Computer Science & Information Technology, Jazan University, Jazan, Saudi Arabia
| | - Ricai Luo
- School of Mathematics and Physics, Hechi University, Yizhou, Guangxi 456300, China
| | - Muhammad Azeem
- Department of Mathematics, Riphah International University Lahore, Pakistan
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11
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Arockiaraj M, Fiona JC, Kavitha SRJ, Shalini AJ, Balasubramanian K. Topological and Spectral Properties of Wavy Zigzag Nanoribbons. MOLECULES (BASEL, SWITZERLAND) 2022; 28:molecules28010152. [PMID: 36615349 PMCID: PMC9822221 DOI: 10.3390/molecules28010152] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/22/2022] [Revised: 12/07/2022] [Accepted: 12/19/2022] [Indexed: 12/28/2022]
Abstract
Low-dimensional graphene-based nanomaterials are interesting due to their cutting-edge electronic and magnetic properties. Their large surface area, strong mechanical resistance, and electronic properties have enabled potential pharmaceutical and opto-electronic applications. Graphene nanoribbons (GNRs) are graphene strips of nanometer size possessing zigzag and armchair edge geometries with tunable widths. Despite the recent developments in the characterization, design and synthesis of GNRs, the study of electronic, magnetic and topological properties, GNRs continue to pose a challenge owing to their multidimensionality. In this study, we obtain the topological and electronic properties of a series of wave-like nanoribbons comprising nanographene units with zigzag-shaped edges. The edge partition techniques based on the convex components are employed to compute the mathematical formulae of molecular descriptors for the wave-like zigzag GNRs. We have also obtained the spectral and energetic properties including HOMO-LUMO gaps, bond delocalization energies, resonance energies, 13C NMR and ESR patterns for the GNRs. All of these computations reveal zero to very low HOMO-LUMO gaps that make these nanoribbons potential candidates for topological spintronics.
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Affiliation(s)
| | - J. Celin Fiona
- Department of Mathematics, Loyola College, Chennai 600034, India
| | | | - Arul Jeya Shalini
- Department of Mathematics, Women’s Christian College, Chennai 600006, India
| | - Krishnan Balasubramanian
- School of Molecular Sciences, Arizona State University, Tempe, AZ 85287-1604, USA
- Correspondence:
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12
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Paull D, Jacob K, Clement J, Arockiaraj M, Paul D, Balasubramanian K. Topological Characterization and Entropy Measures of Tetragonal Zeolite Merlinoites. J Mol Struct 2022. [DOI: 10.1016/j.molstruc.2022.134786] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
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13
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Balasubramanian K. Density Functional and Graph Theory Computations of Vibrational, Electronic and Topological Properties of Porous Nanographenes. J PHYS ORG CHEM 2022. [DOI: 10.1002/poc.4435] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
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14
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Mushtaq S, Arockiaraj M, Fiona JC, Jency J, Balasubramanian K. Topological properties, entropies, stabilities and spectra of armchair versus zigzag coronene-like nanoribbons. Mol Phys 2022. [DOI: 10.1080/00268976.2022.2108518] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/15/2022]
Affiliation(s)
| | | | - J. Celin Fiona
- Department of Mathematics, Loyola College, Chennai, India
| | - Joseph Jency
- Department of Mathematics, Loyola College, Chennai, India
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15
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Topological Study on Triazine-Based Covalent-Organic Frameworks. Symmetry (Basel) 2022. [DOI: 10.3390/sym14081590] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
Abstract
Most of the research has evidenced that there is a strong natural correlation among the chemical properties of molecular structures. This study analyses supramolecular chemistry and investigates topological indices of supramolecular structures called triazine-based covalent-organic frameworks. The use of degree-based topological indices on these chemical molecular structures can aid material scientists in better understanding their chemical and biological properties, thus compensating for the lack of chemical tests. This study aims to theoretically examine the triazine-based covalent-organic frameworks (TriCF) utilizing degree-based topological indices, specifically multiplicative topological indices and entropy measures. A detailed comparison of the computed topological indices of the aforementioned chemical structures is described using graphical depiction.
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16
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Arockiaraj M, Paul D, Klavžar S, Clement J, Tigga S, Balasubramanian K. Relativistic Topological and Spectral Characteristics of Zeolite SAS Structures. J Mol Struct 2022. [DOI: 10.1016/j.molstruc.2022.133854] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/16/2022]
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17
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Rahul M, Clement J, Singh Junias J, Arockiaraj M, Balasubramanian K. Degree-based entropies of graphene, graphyne and graphdiyne using Shannon’s approach. J Mol Struct 2022. [DOI: 10.1016/j.molstruc.2022.132797] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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18
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Arockiaraj M, Jency J, Abraham J, Ruth Julie Kavitha S, Balasubramanian K. Two-dimensional coronene fractal structures: topological entropy measures, energetics, NMR and ESR spectroscopic patterns and existence of isentropic structures. Mol Phys 2022. [DOI: 10.1080/00268976.2022.2079568] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Affiliation(s)
| | - Joseph Jency
- Department of Mathematics, Loyola College, Chennai, India
| | - Jessie Abraham
- Department of Mathematics, KCG College of Technology, Chennai, India
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19
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Lu J, Zhu L, Gao W. Cyclic connectivity index of bipolar fuzzy incidence graph. OPEN CHEM 2022. [DOI: 10.1515/chem-2022-0149] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Abstract
In the performance characterization of chemical molecular structures, several uncertain properties are often encountered, and fuzzy theory is precisely the tool to characterize these uncertainties. When molecular structures are described by molecular graphs, the corresponding fuzzy graph theory is used to characterize the uncertainty of atoms and atomic bonds. In this study, there is introduced cyclic connectivity index and its average version for bipolar fuzzy incidence graph (BFIG), and several theoretical results are obtained in the light of graph theory and fuzzy theory. Finally, the given new fuzzy index is applied to the testing of anti-aging-related drugs yields average uncertainty data for the corresponding molecular structures.
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Affiliation(s)
- Juanjuan Lu
- School of Chemical and Environmental Engineering, Jiangsu University of Technology , Changzhou 213001 , China
| | - Linli Zhu
- School of Computer Engineering, Jiangsu University of Technology , Changzhou 213001 , China
| | - Wei Gao
- School of Information Science and Technology, Yunnan Normal University , Kunming 650500 , China
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20
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Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties. Symmetry (Basel) 2021. [DOI: 10.3390/sym14010034] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023] Open
Abstract
This review article highlights recent developments in symmetry, combinatorics, topology, entropy, chirality, spectroscopy and thermochemistry pertinent to 2D and 1D nanomaterials such as circumscribed-cyclopolyarenes and their heterocyclic analogs, carbon and heteronanotubes and heteronano wires, as well as tessellations of cyclopolyarenes, for example, kekulenes, septulenes and octulenes. We establish that the generalization of Sheehan’s modification of Pólya’s theorem to all irreducible representations of point groups yields robust generating functions for the enumeration of chiral, achiral, position isomers, NMR, multiple quantum NMR and ESR hyperfine patterns. We also show distance, degree and graph entropy based topological measures combined with techniques for distance degree vector sequences, edge and vertex partitions of nanomaterials yield robust and powerful techniques for thermochemistry, bond energies and spectroscopic computations of these species. We have demonstrated the existence of isentropic tessellations of kekulenes which were further studied using combinatorial, topological and spectral techniques. The combinatorial generating functions obtained not only enumerate the chiral and achiral isomers but also aid in the machine construction of various spectroscopic and ESR hyperfine patterns of the nanomaterials that were considered in this review. Combinatorial and topological tools can become an integral part of robust machine learning techniques for rapid computation of the combinatorial library of isomers and their properties of nanomaterials. Future applications to metal organic frameworks and fullerene polymers are pointed out.
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21
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Abstract
Symmetry forms the foundation of combinatorial theories and algorithms of enumeration such as Möbius inversion, Euler totient functions, and the celebrated Pólya’s theory of enumeration under the symmetric group action. As machine learning and artificial intelligence techniques play increasingly important roles in the machine perception of music to image processing that are central to many disciplines, combinatorics, graph theory, and symmetry act as powerful bridges to the developments of algorithms for such varied applications. In this review, we bring together the confluence of music theory and spectroscopy as two primary disciplines to outline several interconnections of combinatorial and symmetry techniques in the development of algorithms for machine generation of musical patterns of the east and west and a variety of spectroscopic signatures of molecules. Combinatorial techniques in conjunction with group theory can be harnessed to generate the musical scales, intensity patterns in ESR spectra, multiple quantum NMR spectra, nuclear spin statistics of both fermions and bosons, colorings of hyperplanes of hypercubes, enumeration of chiral isomers, and vibrational modes of complex systems including supergiant fullerenes, as exemplified by our work on the golden fullerene C150,000. Combinatorial techniques are shown to yield algorithms for the enumeration and construction of musical chords and scales called ragas in music theory, as we exemplify by the machine construction of ragas and machine perception of musical patterns. We also outline the applications of Hadamard matrices and magic squares in the development of algorithms for the generation of balanced-pitch chords. Machine perception of musical, spectroscopic, and symmetry patterns are considered.
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