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A family of chemical compounds known as metal–organic networks (MONs) is composed mainly of clusters of metal ions with organic ligands. It can increase volatility or make substances soluble in organic solvents. By using these salient features, organic compounds generate applications in material sciences for sol–gel processing. A graph’s entropy is utilized as a complexity indicator and is interpreted as the structural information content of the graph. Investigating the entropies of relationship systems is a common occurrence in discrete mathematics, computer science, information theory, statistics, chemistry, and biology. In this article, we investigated the degree-based entropies: geometric arithmetic entropy, atom bond connectivity entropy, general Randic′ entropy, and general sum connectivity entropy for MONs. Furthermore, we created tables for all expressions by using 1–10 values for the ss parameter of these entropies.

For mammals, l-valine, which is a glycogen, is an essential amino acid. A protein made of 20 amino acids, salicylidene and l-valine make the carboxylate ligand which is the base of chiral Schiff. On a large scale, complexes with the ligand are utilized to help in the research work. To locate the exact location of a specific node from all the nodes, the entire node set is developed in a specific manner by choosing a particular subset and this subset is known as the resolving/locating set. This study contributed to the metric dimension of chemical complexes of supramolecular chain in dialkyltin from N-salicylidene-l-valine. We considered the complexes of 2,3,4 and (Cλ⁎)({C}_{\lambda }^{\ast }) chains and proved that the members of resolving sets are highly dependent on the number of vertices.

Graph theory served in different fields of sciences, especially in chemistry in which creating complex structures and studying their enormous properties. Graph operation is a tool to construct complex chemical structures using basic graphs. While studying their properties, topological descriptors are a well-known methodology introduced by chemists, and even after half of a century past, it is still serving. Formally, a topological descriptor or index is a numerical value corresponding to a chemical structure. This numerical value can be easily accessed by a particular equation, for example, the second Zagreb index, the first reformulated Zagreb, and also from the forgotten topological descriptor. In this particular work, we generalized two existing graph operations, and by using these newly developed graph operations, we created two complex structures by using two graph operations, namely, the corona product and double graph operation. Furthermore, to evaluate the chemical properties of these newly generated structures, we used the methodology of topological descriptors, particularly the first and second Zagreb, the first reformulated Zagreb, and forgotten topological descriptors. Moreover, we also presented the closed formulas of the first and second Zagreb co-indices for these newly generated structures.

A big amount of vanadium is used in the industry. It is used as a vanadium carbide stabiliser in steel production. Only $ 0.5\% $ 0.5 % vanadium in an iron alloy will double the tensile strength. Vanadium salts are used in leather, ink, dye manufacturing and yellow ceramic glaze. On the other hand, vanadium pentoxide is used as a catalyst, in the chemical industry, for the production of sulphuric acid and in the petroleum refinery industry. Resolvability parameter is the most discussed topic in graph theory. This helps to reorganise a network in a unique way, sometimes in terms of atoms (metric dimension) and sometimes bound (edge metric dimension). We studied resolvability parameters of vanadium carbide network in this article i.e. metric, edge metric and partition dimension. Among these metrics resolvability defined by parameters taking a subset from the set of vertices, if it has a unique representation with all the vertices, then it is called a resolving set. The number of elements in the smallest resolving set is called the metric dimension. While the number of elements of the minimal subset of vertices taking from vertex set is called the edge metric dimension, if the representations are unique of all edges and the set is called the edge resolving set. This study can have an effect in many ways in the research, particularly, with the help of some key points such as network structure analysis, material science via material properties, graph theory applications, predictive modelling, material testing and characterisation, as well as in industrial applications. [ABSTRACT FROM AUTHOR]

Complex networks are difficult to understand and deal with, yet building a graph of any complex and huge structure, whether chemical or computer-related, is quite straightforward. In the form of a graph theory view, it becomes simple to design networks. Each component's recognition is aided by the ease of its production in the form of graph theory. Finding a vertex (node or atom) in a structure is similarly difficult; in this instance, resolvability parameters come into play. This technique plays a significant role in managing or allowing access to each vertex in relation to some chosen vertices called a resolving set that shapes an entire cluster of vertices into a unique form and every single vertex is easy to recognise or locate, whether the small or large the structure. This article contributes the bounds on the parameter of partition dimension of chemical complexes in dialkyltin of supramolecular chain from N-Salicylidene-L-Valine. We supposed the complexes of chains from 2,3,4 and another chemical network, and we demonstrated that the number of vertices has a significant impact on the composition of resolving partition sets. [ABSTRACT FROM AUTHOR]

A form of porous polymeric material known as metal-organic frameworks is made up of metal ions connected by organic bridging ligands. Entropy is used as a complexity indicator for graphs and is regarded as the graph's structural information content. The graph entropies have evolved into information-theoretic variables for computing the structure information of chemical graphs. The potential use in chemistry of the graph entropy portion has gained the attention of the research community. In this article, we investigate the explicit expressions of graph entropies for two categories of metal organic frameworks (MOFs) named as, $ M_{1}TPyP-M_{2} $ M 1 TPyP − M 2 (TPyP= 5, 10, 15, 20-tetrakis (4-pyridyl) porphyrin and $ M_{1}, M_{2} $ M 1 , M 2 = Fe and Co) MOFs and butylated hydroxytoluene-(BHT-) based metal-organic (M= Co, Fe, Mn, Cr) (MBHT) frameworks. we compute some numeric values for certain graph entropies in Tables. In the last, we present a comprehensive graphic comparison of our new results. [ABSTRACT FROM AUTHOR]

Elliptic curve cryptography has gained attention due to its strong resilience against current cryptanalysis methods. Inspired by the increasing demand for reliable and secure cryptographic methods, our research investigates the relationship between complex mathematical structures and image encryption. A substitution box (S-box) is the single non-linear component of several well-known security systems. Mordell elliptic curves are used because of their special characteristics and the immense computational capacity of Galois fields. These S-boxes are dynamic, which adds a layer of complexity that raises the encryption process's security considerably. We suggest an effective technique for creating S-boxes based on a class of elliptic curves over G F (2 n) , n ≥ 8 . We demonstrate our approach's robustness against a range of cryptographic threats through thorough examination, highlighting its practical applicability. The assessment of resistance of the newly generated S-box to common attack methods including linear, differential, and algebraic attacks involves a thorough analysis. This analysis is conducted by quantifying various metrics such as non-linearity, linear approximation, strict avalanche, bit independence, and differential approximation to gauge the S-box's robustness against these attacks. A recommended method for image encryption involves the use of built-in S-boxes to quickly perform pixel replacement and shuffling. To evaluate the efficiency of the proposed strategy, we employed various tests. The research holds relevance as it can provide alternative guidelines for image encryption, which could have wider consequences for the area of cryptography as a whole. We believe that our findings will contribute to the development of secure communication and data protection, as digital security is becoming increasingly important. [ABSTRACT FROM AUTHOR]

A connected acyclic graph in which the degree of every vertex is at most four is called a molecular tree. A number associated with a molecular tree that can help to approximate the physical or chemical properties of the corresponding molecule is called a topological index. It is of great importance to investigate the relation between the structure and the thermodynamic properties of those molecules. In this paper, we investigated the extreme value of the first reformulated Zagreb index with a given order and degree of a graph. Further, we investigated the molecular trees that attain the maximum and minimum values. As an application, we presented the regression models to predict the acentric factor and entropy of octane isomers. Our extremal graphs give the minimum and the maximum acentric factor and entropy, which satisfied the experimental values. [ABSTRACT FROM AUTHOR]

A T-spherical fuzzy set (TSFS) is an extended and logical algebraic representation to handle uncertainty, with the help of four functions describing four possible aspects of uncertain information. Aczel-Alsina triangular norm (TN) and conorm (TCN) are novel and proved to be more efficient than other existing TNs and TCNs. In our article, we establish the concept of a T-spherical fuzzy Aczel-Alsina graph (TSFAAG). We described the energy of TSFAAG along with the splitting and shadow energy of TSFAAG. Furthermore, we figured out the Randić energy of TSFAAG and obtained some useful results. Moreover, we give the notion of the Aczel-Alsina digraph (TSFAADG). To see the significance of the proposed TSFAADGs, we employed the energy and Randić energy of TSFAADGs for solving the problem of selecting the best investing company by using a decision-making algorithm. The sensitivity analysis of the variable parameters is also discussed and where the effect on ranking results is studied. To see the effectiveness of the proposed work, we did a comparative study and established some remarks. [ABSTRACT FROM AUTHOR]

Topological indices are the fixed numbers associated with the graphs. In recent years, mathematicians used indices to check the pharmacology characteristics and molecular behavior of medicines. In this article the first Zagreb connection number index is computed for the nanotubes VC5C7[ p, q] , HC5C7[ p,q] and Boron triangular Nanotubes. Also, the same index is computed for the Quadrilateral section Pmn$P_{m}^{n}$and Pm+12n$P_{m+\frac{1}{2}}^{n}$cuts from regular hexagonal lattices.

A numerical number associated to the molecular graph G that describes its molecular topology is called topological index. In the study of QSAR and QSPR, topological indices such as atom-bond connectivity index, Randić connectivity index, geometric index, etc. help to predict many physico-chemical properties of the chemical compound under study. Dendrimers are macromolecules and have many applications in chemistry, especially in self-assembly procedures and host-guest reactions. The aim of this report is to compute degree-based topological indices, namely the fourth atom-bond connectivity index and fifth geometric arithmetic index of poly propyl ether imine, zinc porphyrin, and porphyrin dendrimers.

Give us a linked graph, G = (V; E): A vertex w 2 V distinguishes between two components (vertices and edges) x; y 2 E [V if dG(w; x), dG(w; y): Let W1 and W2 be two resolving sets and W1, W2. Then, we can say that the graph G has double resolving set. A nanotube derived from an quadrilateral-octagonal grid belongs to essential and extensively studied compounds in materials science. Nano-structures are very important due to their thickness. In this article, we have discussed the metric dimension of the graphs of nanotubes derived from the quadrilateral-octagonal grid. We proved that the generalized nanotube derived from quadrilateral-octagonal grid have three metric dimension. We also check that the exchange property is also held for this structure. [ABSTRACT FROM AUTHOR]