Alkanes with the First Three Maximal/Minimal Modified First Zagreb Connection Indices

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.

Mathematical analysis and molecular descriptors of two novel metal-organic models with chemical applications

Metal-Organic Networks (MONs) are made by chemical molecules that contain metal ions and organic ligands. A crystalline porous solid called Metal-Organic Networks (MONs) is made up of a [Formula: see text] metal network of ions held in place by a multidentate ligand. (MONs) can be used for gas storage, purification drug delivery, gas separation, catalysis, and sensing applications. There is enormous potential for effective integration and research of MONs in diverse applications. Molecular descriptors are arithmetic measures that reveal a chemical substance's physical and chemical characteristics in its foundational network in a natural relationship. They demonstrate an important role in theoretical and ecological chemistry, and in the field of medicine. In this research, we calculated various recently discovered molecular descriptors viz. the modified version of second zagreb index, harmonic index, reciprocal randic index, modified version of forgotten topological index, redefined first zagreb topological index, redefined second zagreb topological index and redefined third zagreb topological index for two separate metal-organic networks. The numerical and graphical comparative analysis of these considered molecular descriptors are also performed.

Expected value of first Zagreb connection index in random cyclooctatetraene chain, random polyphenyls chain, and random chain network

The Zagreb connection indices are the known topological descriptors of the graphs that are constructed from the connection cardinality (degree of given nodes lying at a distance 2) presented in 1972 to determine the total electron energy of the alternate hydrocarbons. For a long time, these connection indices did not receive much research attention. Ali and Trinajstić [Mol. Inform. 37, Art. No. 1800008, 2018] examined the Zagreb connection indices and found that they compared to basic Zagreb indices and that they provide a finer value for the correlation coefficient for the 13 physico-chemical characteristics of the octane isomers. This article acquires the formulae of expected values of the first Zagreb connection index of a random cyclooctatetraene chain, a random polyphenyls chain, and a random chain network with l number of octagons, hexagons, and pentagons, respectively. The article presents extreme and average values of all the above random chains concerning a set of special chains, including the meta-chain, the ortho-chain, and the para-chain.

Topological Characterizations of Crystal Cubic Carbon Structures

Graph theory (GT) serves as a mathematical foundation which helps to manipulate, develop, analyze and comprehend the chemical networks or structures and their characteristics. Molecular graph is a graph made up of vertices (atoms) and edges (chemical bonds between atoms). Chemical GT applied geometrical and combinatorial GT to model the important chemical structures of chemistry. Chemical GT has a wide range of utilizations in the study of chemical structures. The examination and manipulation of chemical structural information is made feasible by utilizing the numerical graph invariants. A graph invariant or a topological index (TI) is a numerical measure of a chemical compound which is capable to describe the properties such as, melting point, freezing point, density, pressure, tension, temperature, etc., of chemical compounds. In this manuscript, we compute connection based Zagreb indices (ZIs), namely, first connection based Zagreb index (CBZI), second CBZI, modified first CBZI, modified second CBZI and modified third CBZI for one of the significant type of molecular structure named as crystal cubic carbon structure which is the most important allotrope of carbon atom. Moreover, to examine the superiority and authenticity of our computed TIs, we compare the calculated values of these ZIs for crystal cubic carbon structure with each other. This comparison enable us to check that which CBZI is more authentic and superior to predict the physical and chemical properties of crystal cubic carbon structure.

Study of Transformed ηζ Networks via Zagreb Connection Indices

A graph is a tool for designing a system’s required interconnection network. The topology of such networks determines their compatibility. For the first time, in this work we construct subdivided ηζ network S(ηζΓ) and discussed their topology. In graph theory, there are a variety of invariants to study the topology of a network, but topological indices are designed in such a way that these may transform the graph into a numeric value. In this work, we study S(ηζΓ) via Zagreb connection indices. Due to their predictive potential for enthalpy, entropy, and acentric factor, these indices gain value in the field of chemical graph theory in a very short time. ηζΓ formed by ζ time repeated process which consists ηζ copies of graph Γ along with η2|V(Γ)|ζηζ−1 edges which used to join these ηζ copies of Γ. The free hand to choose the initial graph Γ for desired network S(ηζΓ) and its relation with chemical networks along with the repute of Zagreb connection indices enhance the worth of this study. These computations are theoretically innovative and aid topological characterization of S(ηζΓ).

Computing Connection-Based Topological Indices of Dendrimers

Dendrimers are artificially synthesized polymeric macromolecules composed of frequently branching chains called monomers. Topological indices (TIs) are the molecular descriptors which characterize the topology and help to correlate the distinct psychochemical properties such as stability, boiling point, and strain energy of molecular compounds. TIs are classified on the basis of their degrees, distance, and spectrum. Among these TIs, connection-based topological descriptors have great significance. In this study, we initiate the general expressions to compute multiplicative connection Zagreb indices (MZIs), named as first MZCI, second MZCI, third MZCI, fourth MZCI, modified first MZCI, modified second MZCI, and modified third MZCI of two exceptional dendrimers nanostars, namely, poly (propyl) ether imine (PPIE) dendrimer and polypropylenimine octamin (PPIO) dendrimer. Furthermore, in order to check the superiority of our computed results, a comparative analysis is conducted.

Study of θϕ Networks via Zagreb Connection Indices

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θϕ formed by ϕ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θϕΩ. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of θϕΩ to enable its emerging use.

Molecular Descriptor Analysis of Certain Isomeric Natural Polymers

Polymers, drugs, and almost all chemical or biochemical compounds are frequently modeled as diverse $\omega$ -cyclic, acyclic, bipartite, and polygonal shapes and regular graphs. Molecular descriptors (topological indices) are the numerical quantities and computed from the molecular graph $\mathrm{\Gamma }$ (2D lattice). These descriptors are highly significant in quantitative structure-property or activity relationship (QSPR and QSAR) modeling that provides the theoretical and the optimal basis to expensive experimental drug design. In this paper, we study three isomeric natural polymers of glucose (polysaccharides), namely, cellulose, glycogen, and amylopectin (starch), having promising pharmaceutical applications, exceptional properties, and fascinating molecular structures. We intend to investigate and compute various closed-form formulas such as $\text{ABC}$ , $\text{GA}$ , sum-connectivity ${\chi }_{\left(-1/2\right)}$ , ${\text{ABC}}_4$ , ${\text{GA}}_5$ , and Sanskruti indices for the aforementioned macromolecules. Also, we present the closed-form formulas for the first, second, modified, and augmented Zagreb indices, inverse and general Randić indices, and symmetric division deg, harmonic, and inverse sum indices. Furthermore, we provide a comparative analysis using 3D graphs for these families of macromolecules to clarify their nature.

Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs

Representation or coding of the molecular graphs with the help of numerical numbers plays a vital role in the studies of physicochemical and structural properties of the chemical compounds that are involved in the molecular graphs. For the first time, the modified first Zagreb connection index appeared in the paper by Gutman and Trinajstic (1972) to compute total electron energy of the alternant hydrocarbons, but after that, for a long time, it has not been studied. Recently, Ali and Trinajstic (2018) restudied the first Zagreb connection index ZC1, the second Zagreb connection index ZC2, and the modified first Zagreb connection index ZC1∗ to find entropy and acentric factor of the octane isomers. They also reported that the values provided by the International Academy of Mathematical Chemistry show better chemical capability of the Zagreb connection indices than the ordinary Zagreb indices. Assume that S1 and S2 denote the operations of subdivision and semitotal point, respectively. Then, the S-sum graphs Q1+QS2 are obtained by the cartesian product of SQ1 and Q2, where S∈S1,S2, Q1andQ2 are any connected graphs, and SQ1 is a graph obtained after applying the operation S on Q1. In this paper, we compute the Zagreb connection indices (ZC1, ZC2, and ZC1∗) of the S-sum graphs in terms of various topological indices of their factor graphs. At the end, as an application of the computed results, the Zagreb connection indices of the S-sum graphs obtained by the particular classes of alkanes are also included.