Applications of magnesium iodide structure via modified-polynomials.

A relatively recent approach in molecular graph theory for analyzing chemical networks and structures is called a modified polynomial. It emphasizes the characteristics of molecules through the use of a polynomial-based procedure and presents numerical descriptors in algebraic form. The Quantitative Structure-Property Relationship study makes use of Modified Polynomials (M-Polynomials) as a mathematical tool. M-Polynomials used to create connections between a material's various properties and its structural characteristics. In this study, we calculated several modified polynomials and gave a polynomial description of the magnesium iodide structure. Particularly, we computed first, second and modified Zagreb indices based M-polynomials. Randic index, and inverse Randic indices based M-polynomials are also computed in this work.

Study of some graph theoretical parameters for the structures of anticancer drugs.

Eigenvalues have great importance in the field of mathematics, and their relevance extends beyond this area to include several other disciplines such as economics, chemistry, and numerous fields. According to our study, eigenvalues are utilized in chemistry to express a chemical compound's numerous physical properties as well as its energy form. It is important to get a comprehensive understanding of the interrelationship underlying mathematics and chemistry. The anti-bonding phase is correlated with positive eigenvalues, whereas the bonding level is connected with negative eigenvalues. Additionally, the non-bonded level corresponds to eigenvalues of zero. This study focuses on the analysis of various structures of anticancer drugs, specifically examining their characteristic polynomials, eigenvalues of the adjacency matrix, matching number and nullity. Consequently, the selected structures of the aforementioned anticancer drugs exhibit stability since they are composed of closed-shell molecules, characterized by a nullity value of zero.

Analysis and simulation study of the HIV/AIDS model using the real cases.

We construct a model to investigate HIV/AIDS dynamics in real cases and study its mathematical analysis. The study examines the qualitative outcomes and confirms the local and global asymptotic stability of both the endemic equilibrium and the disease-free equilibrium. The model's criteria for exhibiting both local and global asymptotically stable behavior are examined. We compute the endemic equilibria and obtain the existence of a unique positive endemic equilibrium. The data is fitted to the model using the idea of nonlinear least-squares fitting. Accurate parameter values are achieved by fitting the data to the model using a 95% confidence interval. The basic reproduction number is computed using parameters that have been fitted or estimated. Sensitivity analysis is performed to discover the influential parameters that impact the reproduction number and the eradication of the disease. The results show that implementing preventive measures can reduce HIV/AIDS cases.

Valency-based structural properties of gamma-sheet of boron clusters.

Boron cluster sheets are two-dimensional boron atom-based formations called borophene. They are similar to the two-dimensional sheet known as graphene, which is composed of carbon atoms arranged in a hexagonal lattice. The unique electrical, mechanical, and thermal properties of borophene make it a sought-after substance for a variety of uses, such as catalysis, energy storage, and electronics. There are two ways to manufacture borophene: chemical vapor deposition and molecular beam epitaxy. Vertex-edge valency-based topological descriptors are a great example of a molecular descriptor that provides information on the connection of atoms in a molecule. These descriptions are based on the notion that a node's value in a molecular network is the sum of the valency of those atoms that are directly connected to that node. In this article, we discussed some novel vertex-edge (ve) and edge-vertex (ev) topological descriptors and found their formulations for the boron cluster or borophene sheets. Also, we show the numerical and graphical comparison of these descriptors in this article.

Topological Insights into Nanostar Dendrimers by Computing the Augmented Zagreb Index.

BACKGROUND: The field of nanobiotechnology uses precise nanofabrication techniques to advance our understanding and control of biological systems. Due to their remarkable properties, dendrimers, which are hyperbranched macromolecular structures with distinct and well-defined architectures, have emerged as pivotal entities within this field. They are gaining increasing attention for their potential to catalyze a paradigm shift in medical therapeutics, biotechnological applications, and advanced material sciences. OBJECTIVE: This paper focuses on a novel analytical expression and determines the precise value of the augmented Zagreb index, a topological descriptor, for eight classes of nanostar dendrimers. METHODS: The Zagreb index is a topological invariant to predict molecular behaviour and reactivity. In this paper, we have explored its application in characterizing the branching of nanostar dendrimers through computational modelling and mathematical rigor. RESULTS: Our research has measured the augmented Zagreb index for nanostar dendrimers, which fall into eight distinct classes. The results better explain the relationship between the dendrimers' topology and chemical properties. This correlation has implications for their structural stability and reactivity, potentially leading to new applications. CONCLUSION: Developing the augmented Zagreb index for nanostar dendrimers is a significant breakthrough in nanobiotechnology. Based on the correlation between the calculated topological index and the corresponding molecular attributes, our analytical approach has opened up new possibilities for designing and synthesizing dendrimers tailored to specific functions in medical and material science applications. This precise topological quantification could significantly enhance the utility and functionalization of dendrimers in cutting-edge nanotechnological applications.

Edge based metric dimension of various coffee compounds.

An important dietary source of physiologically active compounds, coffee also contains phenolic acids, diterpenes, and caffeine. According to a certain study, some coffee secondary metabolites may advantageously modify a number of anti-cancer defense systems. This research looked at a few coffee chemical structures in terms of edge locating numbers or edge metric size to better understand the mechanics of coffee molecules. Additionally, this research includes graph theoretical properties of coffee chemical structures. The chemicals found in coffee, such as caffeine, diterpene or cafestol, kahweol, chlorogenic, caffeic, gallotannins, and ellagitannins, are especially examined in these publications.

Fundamental aspects of the molecular topology of fuchsine acid dye with connection numbers.

Fuchsine acid serves as a supramolecular dye in Masson's trichrome stain, finding extensive applications in histology. It is also utilized with picric acid in Van Gieson's method to reveal red collagen fibers and in Masson's trichrome to highlight smooth muscle in contrast to collagen. Beyond these applications, it plays a crucial role in electronic fields and photonic devices as an organic semiconductor. Therefore, investigating and predicting the complex molecular structure of fuchsine acid becomes essential, serving as the foundation for understanding its physicochemical features. This article employs topological modeling, specifically a connection number edge partition, to explore the supramolecular nature of fuchsine acid. Closed formulae for key degree-based molecular descriptors are derived, aiming to illuminate the effectiveness of these descriptors for QSAR and QSPR analyses.

Double edge resolving set and exchange property for nanosheet structure.

The exploration of edge metric dimension and its applications has been an ongoing discussion, particularly in the context of nanosheet graphs formed from the octagonal grid. Edge metric dimension is a concept that involves uniquely identifying the entire edge set of a structure with a selected subset from the vertex set, known as the edge resolving set. Let's consider two distinct edge resolving sets, denoted as Re1 and Re2, where Re1≠Re2. In such instances, it indicates that the graph G possesses a double-edge resolving set. This implies the existence of two different subsets of the vertex set, each capable of uniquely identifying the entire edge set of the graph. In this article, we delve into the edge metric dimension of nanosheet graphs derived from the octagonal grid. Additionally, we initiate a discussion on the exchange property associated with the edge resolving set. The exchange property holds significance in the study of resolving sets, playing a crucial role in comprehending the structure and properties of the underlying graph.

Connection number topological aspect for backbone DNA networks.

The present study investigates the complex topological characteristics of DNA networks, with a specific emphasis on the innovative metric known as Connection Number (CN) as a key factor in determining network structure. The Connection Number, represented as CN(v) for a vertex v, measures the count of unique paths that link v to every other vertex in the network. By employing rigorous mathematical modeling and analysis techniques, we are able to reveal the profound implications of CN (complex networks) in characterizing the structural robustness and dynamics of information flow within DNA networks. The study of how the theory of graphs and chemicals interact is known as chemical graph theory. This paper, computing the hyper Zagreb connection index, augmented connection index, inverse sum connection index, harmonic connection index, symmetric division connection index, geometric arithmetic connection index, and atom bond connectivity connection index, of two significant types of backbone DNA and Barycentric subdivision of backbone DNA networks. Direct method computation is used to produce these Connection-based topological descriptors.

Some stable and closed-shell structures of anticancer drugs by graph theoretical parameters.

The eigenvalues are significant in mathematics, but they are also relevant in other domains like as chemistry, economics, and a variety of others. In terms of our research, eigenvalues are used in chemistry to represent not only the form of energy but also the various physicochemical aspects of a chemical substance. We must comprehend the connection between mathematics and chemistry. The antibonding level is related to positive eigenvalues, the bonding level is associated to negative eigenvalues, and the nonbonding level is linked to zero eigenvalues. In this work, we studied some anticancer drug structures in terms of nullity, matching number, eigenvalues of adjacency matrix, and characteristics polynomials. As a result, Carmustine, Caulibugulone-E, Aspidostomide-E anticancer drug structures are stable, closed-shell molecules since their nullity is equal to zero.

Partition dimension of COVID antiviral drug structures.

In November 2019, there was the first case of COVID-19 (Coronavirus) recorded, and up to 3$ ^{rd }$ of April 2020, 1,116,643 confirmed positive cases, and around 59,158 dying were recorded. Novel antiviral structures of the SARS-COV-2 virus is discussed in terms of the metric basis of their molecular graph. These structures are named arbidol, chloroquine, hydroxy-chloroquine, thalidomide, and theaflavin. Partition dimension or partition metric basis is a concept in which the whole vertex set of a structure is uniquely identified by developing proper subsets of the entire vertex set and named as partition resolving set. By this concept of vertex-metric resolvability of COVID-19 antiviral drug structures are uniquely identified and helps to study the structural properties of structure.