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Hui W, Aslam A, Kanwal S, Akram M, Shaikh TS, Zuo X. Ordering Unicyclic Connected Graphs with Girth g ≥ 3 Having Greatest SK Indices. Symmetry (Basel) 2023. [DOI: 10.3390/sym15040871] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/08/2023] Open
Abstract
For a graph, the SK index is equal to the half of the sum of the degrees of the vertices, the SK1 index is equal to the half of the product of the degrees of the vertices, and the SK2 index is equal to the half of the square of the sum of the degrees of the vertices. This paper shows a simple and unified approach to the greatest SK indices for unicyclic graphs by using some transformations and characterizes these graphs with the first, second, and third SK indices having order r ≥ 5 and girth g ≥ 3, where girth is the length of the shortest cycle in a graph.
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Affiliation(s)
- Wang Hui
- General Education Department, Anhui Xinhua University, Hefei 230088, China
| | - Adnan Aslam
- Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore 54000, Pakistan
| | - Salma Kanwal
- Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan
| | - Mahnoor Akram
- Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan
| | - Tahira Sumbal Shaikh
- Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan
| | - Xuewu Zuo
- General Education Department, Anhui Xinhua University, Hefei 230088, China
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Baby A, Julietraja K, Xavier DA. On Molecular Structural Characterization of Cyclen Cored Dendrimers. Polycycl Aromat Compd 2023. [DOI: 10.1080/10406638.2023.2179641] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/25/2023]
Affiliation(s)
- Annmaria Baby
- Department of Mathematics, Loyola College, University of Madras, Chennai, India
| | - K. Julietraja
- Department of Mathematics, St. Joseph’s College of Engineering, Chennai, India
| | - D. Antony Xavier
- Department of Mathematics, Loyola College, University of Madras, Chennai, India
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Liu JB, Xavier DA, Varghese ES, Baby A, Mathew D. Molecular Descriptors of Porphyrin-based Dendrimer. Polycycl Aromat Compd 2022. [DOI: 10.1080/10406638.2022.2112714] [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)
- Jia-Bao Liu
- School of Mathematics and Physics, Anhui Jianzhu University, Hefei, China
| | - D. Antony Xavier
- Department of Mathematics, Loyola College (Affiliated to the University of Madras), Chennai, India
| | - Eddith Sarah Varghese
- Department of Mathematics, Loyola College (Affiliated to the University of Madras), Chennai, India
| | - Annmaria Baby
- Department of Mathematics, Loyola College (Affiliated to the University of Madras), Chennai, India
| | - Deepa Mathew
- Department of Mathematics, St. Joseph’s College, Bangalore, India
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Xavier DA, Varghese ES, Baby A, Mathew D, K. A. Kaabar M. Distance based topological descriptors of Zinc Porphyrin Dendrimer. J Mol Struct 2022. [DOI: 10.1016/j.molstruc.2022.133614] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Chu YM, Numan M, Butt SI, Siddiqui MK, Ullah R, Cancan M, Ali U. Degree-Based Topological Aspects of Polyphenylene Nanostructures. Polycycl Aromat Compd 2022. [DOI: 10.1080/10406638.2020.1852271] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- Yu-Ming Chu
- Department of Mathematics, Huzhou University, Huzhou, P. R. China
- Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha, P. R. China
| | - Muhammad Numan
- Department of Mathematics, COMSATS University Islamabad, Attock, Pakistan
| | - Saad Ihsan Butt
- Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan
| | | | - Rizwan Ullah
- Department of Mathematics, Islamia College, Peshawar, Pakistan
| | - Murat Cancan
- Faculty of Education, Van Yuzuncu Yil University, Van, Turkey
| | - Usman Ali
- Department of Mathematics, COMSATS University Islamabad, Attock, Pakistan
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Abstract
Topological indices are numerical parameters used to study the physical and chemical residences of compounds. Degree-based topological indices have been studied extensively and can be correlated with many properties of the understudy compounds. In the factors of degree-based topological indices, M-polynomial played an important role. In this paper, we derived closed formulas for some well-known degree-based topological indices like first and second Zagreb indices, the modified Zagreb index, the symmetric division index, the harmonic index, the Randić index and inverse Randić index, and the augmented Zagreb index using calculus.
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Computing Zagreb and Randić Indices of PEG-Cored Dendrimers Used for Drug and Gene Delivery. J CHEM-NY 2020. [DOI: 10.1155/2020/2821510] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
Zagreb and Randić indices are the most commonly used degree-based topological indices in the study of drug design and development. In molecular topology, M-polynomials are also used to calculate the degree-based topological indices of chemical structures. In this paper, we derive the M-polynomials for the PEG-cored PAMAM, carbosilane, and poly (lysine) dendrimers and calculate their first, second, and second modified Zagreb indices and the Randić index.
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Chu ZQ, Salman M, Munir A, Khalid I, Rehman MU, Liu JB, Ali F. Some topological indices of dendrimers determined by their Banhatti polynomials. HETEROCYCL COMMUN 2020. [DOI: 10.1515/hc-2020-0102] [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
AbstractSeveral properties of chemical compounds in a molecular structure can be determined with the aid of mathematical languages provided by various types of topological indices. In this paper, we consider eight dendrimer structures in the context of valency based topological indices. We define four Banhatti polynomials for general (molecular) graphs, and compute them for underline dendrimers. We use these polynomials to determine four Banhatti indices. We also determine Zagreb (first, second and hyper) and forgotten indices by developing their relationships with Banhatti indices.
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Affiliation(s)
- Zheng-Qing Chu
- General Education Department, Anhui Xinhua University, Hefei, 230088, P.R. China
| | - Muhammad Salman
- Department of Mathematics, The Islamia University of Bahawalpur, 63100, BahawalpurPakistan
| | - Asia Munir
- Department of Mathematics, The Islamia University of Bahawalpur, 63100, BahawalpurPakistan
| | - Imran Khalid
- Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, 60800, MultanPakistan
| | - Masood Ur Rehman
- Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, 22500, Pakistan
| | - Jia-Bao Liu
- School of Mathematics and Physics, Anhui Jianzhu University, Hefei, Anhui, 230601, P.R.China
| | - Faisal Ali
- Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, 60800, MultanPakistan
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Jahanbani A, Shao Z, Sheikholeslami SM. Calculating degree based multiplicative topological indices of Hyaluronic Acid-Paclitaxel conjugates’ molecular structure in cancer treatment. J Biomol Struct Dyn 2020; 39:5304-5313. [DOI: 10.1080/07391102.2020.1800512] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Affiliation(s)
- Akbar Jahanbani
- Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
| | - Zehui Shao
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
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Shao Z, Jahanbani A, Sheikholeslami SM. Multiplicative Topological Indices of Molecular Structure in Anticancer Drugs. Polycycl Aromat Compd 2020. [DOI: 10.1080/10406638.2020.1743329] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Affiliation(s)
- Zehui Shao
- Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
| | - Akbar Jahanbani
- Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran
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Abstract
AbstractDendrimers have an incredibly strong potential because their structure allows multivalent frameworks, i.e. one dendrimer molecule has many possible destinations to couple to a functioning species. Researchers expected to utilize the hydrophobic conditions of the dendritic media to lead photochemical responses that make the things that are artificially tested. Carboxylic acid and phenol- terminated water-dissolvable dendrimers were joined to set up their utility in tranquilize conveyance and furthermore driving compound reactions in their inner parts. This may empower scientists to associate both concentrating on atoms and medication particles to the equivalent dendrimer, which could diminish negative manifestations of prescriptions on sound and health cells. Topological indices are numerical numbers associated with the graphs of dendrimers and are invariant up to graph isomorphism. These numbers compare certain physicochemical properties like boiling point, strain energy, stability, etc. of a synthetic compound. There are three main types of topological indices, i.e degree-based, distance-based and spectrum-based. In this paper, our aim is to compute some degree-based indices and polynomials for some dendrimers and polyomino chains. We computed redefined first, second and third Zagreb indices of PAMAM dendrimers PD1, PD2, and DS1 and linear Polyomino chain Ln , Zigzag Polyomino chain Zn, polyomino chain with n squares and of m segments $B_{n}^{1}$and $B_{n}^{2}$We also computed some Zagreb polynomials of understudy dendrimers and chains.
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Abstract
There is extremely a great deal of mathematics associated with electrical and electronic engineering. It relies upon what zone of electrical and electronic engineering; for instance, there is much increasingly theoretical mathematics in communication theory, signal processing and networking, and so forth. Systems include hubs speaking with one another. A great deal of PCs connected together structure a system. Mobile phone clients structure a network. Networking includes the investigation of the most ideal method for executing a system. Graph theory has discovered a significant use in this zone of research. In this paper, we stretch out this examination to interconnection systems. Hierarchical interconnection systems (HINs) give a system to planning systems with diminished connection cost by exploiting the area of correspondence that exists in parallel applications. HINs utilize numerous levels. Lower-level systems give nearby correspondence, while more significant level systems encourage remote correspondence. HINs provide issue resilience within the sight of some defective nodes and additionally interfaces. Existing HINs can be comprehensively characterized into two classes: those that use nodes or potential interface replication and those that utilize reserve interface nodes.
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Irregularity Indices of Dendrimer Structures Used as Molecular Disrupter in QSAR Study. J CHEM-NY 2019. [DOI: 10.1155/2019/5371254] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Dendrimers are rising polymeric structures known for their flexibility in medication conveyance and high usefulness, whose properties are same biomolecules. These nanostructured macromolecules have shown potential capacities in capturing as well as conjugating the high subatomic weight hydrophilic/hydrophobic substances by host-visitor collaborations and covalent holding (prodrug approach) individually. In quantitative structure-property relationships (QSPR), topological indices are utilized to gather properties of dendrimers. Topological indices catch symmetry of dendrimer structures and give it a logical reasoning to predict properties, for instance, viscosity, boiling points, the radius of gyrations, etc. In this report, we intend to examine dendrimers through irregularity indices that are valuable in QSPR studies. We studied sixteen irregularity indices if diverse dendrimer structures.
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Computations of the M-Polynomials and Degree-Based Topological Indices for Dendrimers and Polyomino Chains. Int J Anal Chem 2018; 2018:1709073. [PMID: 30515215 PMCID: PMC6236665 DOI: 10.1155/2018/1709073] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2018] [Accepted: 10/18/2018] [Indexed: 11/18/2022] Open
Abstract
Topological indices correlate certain physicochemical properties like boiling point, stability, and strain energy of chemical compounds. In this report, we compute M-polynomials for PAMAM dendrimers and polyomino chains. Moreover, by applying calculus, we compute nine important topological indices of under-study dendrimers and chains.
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Some Reverse Degree-Based Topological Indices and Polynomials of Dendrimers. MATHEMATICS 2018. [DOI: 10.3390/math6100214] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Topological indices collect information from the graph of molecule and help to predict properties of the underlying molecule. Zagreb indices are among the most studied topological indices due to their applications in chemistry. In this paper, we compute first and second reverse Zagreb indices, reverse hyper-Zagreb indices and their polynomials of Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly (ethylene amido amine) dendrimers.
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