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.

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.

A medical image encryption scheme based on Mobius transformation and Galois field.

Data security and privacy are considered to be the biggest problems faced by service providers who have worked with public data for a long time. A key element of modern encryption that is utilized to increase textual confusion is the Substitution box (S-box) and the algebraic strength of the S-box has a significant impact on how secure the encryption method is. In this article, we present a unique method that uses a linear fractional transformation on a finite field to produce cryptographically robust S-boxes. Firstly, we choose a specific irreducible polynomial of degree 8 in Z2[x] to construct GF(28). Later, we used the action of PGL(2,GF(28)) on GF(28) to generate a robust S-box. The effectiveness of the built-in S-box was evaluated using several criteria including non-linearity, differential uniformity, strict avalanche criteria, linear approximation probability, and bit independence criterion. The proposed S-box's characteristics are compared to those of most recent S-boxes to confirm the higher performance. Additionally, the S box was used to encrypt images to show its usefulness for multimedia security applications. We performed several tests, including contrast, correlation, homogeneity, entropy, and energy, to evaluate the success of the encryption technique. The proposed method for ciphering an image is very effective, as proven by its comparison with several S boxes.

Sombor topological indices for different nanostructures.

Euclidean geometry is utilized to establish the Sombor graph parameter and its invariants. It is sum of all adjacent vertices in graph theory dϒ2+dΓ2 where dϒ is the degree of the vertex ϒ. Geometrical interpretation is used to describe the new Sombor indices types. We examined, recently developed Sombor indices for various nanotube Y-junctions in this article. In specifically, the first area-based Sombor index was introduced by Euclidean geometry. Angular orientation concept to construct the second, fourth, and sixth Sombor graph parameters, while third and fifth Sombor graph parameters are constructed by perimeter.

Edge valency-based entropies of tetrahedral sheets of clay minerals.

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.

Comparative study of five topological invariants of supramolecular chain of different complexes of N-salicylidene-L-valine.

L-valine is a crucial amino acid that has rising market demand and numerous uses. It can be used to make specific nutrients, animal feed additives, cosmetic ingredients, and other things in the medical and agricultural fields. N-salicylidene-L-valine (NsLv) is attracting a lot of attention due to its unusual structure and enhanced catalytic and cytotoxic activities. Topological index is a numerical value which is associated with the molecular structure. It is very helpful to predict physio-chemical properties and Quantitative structure-activity relationship and Quantitative structure-property relationship modeling. We study the supramolecular chain (Sc) in the dialkyl tin of complexes 2, 3 and 4 of NsLv to better understand this structure and its topological index-related characteristics. Additionally, we compare topological indices and analyze how these structures relate to one another using concrete examples.

Some new results on the face index of certain polycyclic chemical networks.

Silicate minerals make up the majority of the earth's crust and account for almost 92 percent of the total. Silicate sheets, often known as silicate networks, are characterised as definite connectivity parallel designs. A key idea in studying different generalised classes of graphs in terms of planarity is the face of the graph. It plays a significant role in the embedding of graphs as well. Face index is a recently created parameter that is based on the data from a graph's faces. The current draft is utilizing a newly established face index, to study different silicate networks. It consists of a generalized chain of silicate, silicate sheet, silicate network, carbon sheet, polyhedron generalized sheet, and also triangular honeycomb network. This study will help to understand the structural properties of chemical networks because the face index is more generalized than vertex degree based topological descriptors.

Tetrahedral sheets of clay minerals and their edge valency-based entropy measures.

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.

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.

Extremal (n,m)-Graphs w.r.t General Multiplicative Zagreb Indices.

BACKGROUND: A topological index of a molecular graph is the numeric quantity which can predict certain physical and chemical properties of the corresponding molecule. Xu et al. introduced some graph transformations which increase or decrease the first and second multiplicative Zagreb indices and proposed a unified approach to characterize extremal (n, m)- graphs. METHOD: Graph transformations are used to find the extremal graphs, these transformations either increase or decrease the general multiplicative Zagreb indices. By applying the transformations which increase the general multiplicative Zagreb indices we find the graphs with maximal general multiplicative Zagreb indices and for minimal general Zagreb indices we use the transformations which decrease the index. RESULT: In this paper, we extend the Xu's results and show that the same graph transformations increase or decrease the first and second general multiplicative Zagreb indices for . As an application, the extremal acyclic, unicyclic and bicyclic graphs are presented for general multiplicative Zagreb indices. CONCLUSION: By applying the transformation we investigated that in the class of acyclic, unicyclic and bicyclic graphs, which graph gives the minimum and the maximum general multiplicative Zagreb indices.

Comparative study of vertex-edge based indices for semi-capped carbon nanotubes.

Manufacturing relatively inexpensive items in every area of engineering and science is the major focus of exploration resultant the world's contemporary economic setback. Making small-sized items that are inexpensive and lightweight while providing high quality is critical in today's and tomorrow's worlds. Nanotechnology has a significant role to play in this situation. Nano-objects or, in general, nanomaterials are especially preferred; nanotubes, especially those comprised of carbon, are one of the most popular types of nanostructures, and they are applied in a variety of chemical, biological and technical applications. This notion prompted us to investigate their many physical and chemical characteristics. We utilized topological descriptors to evaluate diverse nanotube structures such as armchair carbon and semi-capped nanotubes by using vertex-edge based indices to characterize distinct chemical structures via numerical quantitative analysis. Furthermore, we examined uncapped and semi-capped armchair carbon nanotubes and achieved adequate comparative findings.

Characteristics studies of molecular structures in drugs.

In theoretical medicine, topological indices are defined to test the medicine and pharmacy characteristics, such as melting point, boiling point, toxicity and other biological activities. As basic molecular structures, hexagonal jagged-rectangle and distance-regular structure are widely appeared in medicine, pharmacy and biology engineering. In this paper, we study the chemical properties of hexagonal jagged-rectangle from the mathematical point of view. Several vertex distance-based indices are determined. Furthermore, the Wiener related indices of distance-regular structure are also considered.

Forgotten topological index of chemical structure in drugs.

A massive of early drug tests implies that there exist strong inner relationships between the bio-medical and pharmacology characteristics of drugs and their molecular structures. The forgotten topological index was defined to be used in the analysis of drug molecular structures, which is quite helpful for pharmaceutical and medical scientists to grasp the biological and chemical characteristics of new drugs. Such tricks are popularly employed in developing countries where enough money is lacked to afford the relevant chemical reagents and equipment. In our article, by means of drug molecular structure analysis and edge dividing technology, we present the forgotten topological index of several widely used chemical structures which often appear in drug molecular graphs.