Abstract Mathematical chemistry is the study of a chemical substance’s molecular structure as a graph and the use of computational methods and graph theory to mathematical problems. One important tool in this field that gives a network structure a numerical value is the topological index. It can be thought of as a mathematical process that gives a chemical compound’s physico-chemical properties a numerical number. R topological indices are the most recent indices that are based on the sum and multiplication of degrees in a neighborhood. In this work, we first investigate the prediction ability of the R indices for octane isomers in order to assess their potential usefulness. R indices of octanes were found to be highly correlated with entropy and acentric factor characteristics. Furthermore, stronger associations between R indices and the Randic, Wiener, and Zagreb indices of octanes were found. These results suggest that R indices will play a significant role in QSPR research.

Abstract A new coordination polymer [Cd 2 (dpa) 2 (bbib) 2 ] n ·2 n H 2 O ( 1 ), (bbib = 1,4-bis(1 H -benzimidazolyl)butane, H 2 dpa = 2,2′-diphenit acid) has been successfully prepared under hydrothermal conditions. The structure of 1 belongs to a 2D network structure, which further stacks together to form a 3D supramolecular architecture through π–π and C–H⋯π interactions. Additionally, the solid-state fluorescence spectrum showed that 1 had a maximum emission wavelength of 427 nm at an excitation wavelength of 350 nm. In the fluorescence sensing experiment, 1 showed selective recognition ability to Fe 3+ with a detection limit of 0.16 μmol·L −1 . Furthermore, quantum chemical calculations were performed on the “molecular fragments” that were derived from the crystal structures of 1 . These calculations utilized the PBE0/LANL2DZ method and were executed through the Gaussian16 program. The calculated results demonstrate a substantial covalent interaction existing between the coordination atoms and the Cd( ii ) ions.

Abstract Topological indices (TIs), as numerical descriptors derived from molecular graphs, offer critical insights into structural properties of chemical compounds by quantifying atomic connectivity, independent of spatial configuration. These indices are pivotal in quantitative structure–activity relationship studies, enabling the prediction of physicochemical and biological behaviors of compounds prior to experimental analysis. In this study, we focus on the computational application of novel Sombor indices to key main group metal-based systems, including bismuth(iii) iodide – a prominent heavy main group metal halide – as well as structurally related TOXs and silicates, which underscore the versatility of main group elements in forming diverse architectures. Additionally, we investigate nanostar dendrimers to explore topological trends in complex, hyperbranched frameworks relevant to main group metalloid-containing polymers. To validate the broader chemical utility of these indices, we analyze their degeneracy and discriminative capacity across a series of octane isomers, establishing a comparative foundation for their efficacy in distinguishing structural nuances. Our results demonstrate that these TIs exhibit strong correlations with enthalpy of vaporization, standard enthalpy of vaporization, acentric factor, and density. All evaluated indices show identical degeneracy rates of 78.67%, underscoring their consistency in structural analysis. These findings highlight the potential of Sombor indices as robust tools for modeling structure–property relationships for designing advanced materials and catalytic systems. This work bridges computational chemistry with applied main group metal research, offering predictive strategies that could streamline the development of main group derivatives for industrial applications.

Abstract New Schiff base of 5-amino-1,3-diphenyl-1H-pyrazole with 5-bromo salicylaldehyde and its cobalt and nickel complexes have been synthesized by condensation reaction. The synthesized compounds were characterized by numerous spectral (UV–vis, IR, 1H and 13C NMR, LC–MS) studies and docking studies. A square planar geometry for Co(iii) and Ni(ii) complexes has been proposed. The infrared spectral results confirm the involvement of azomethine nitrogen and phenolic oxygen in the coordination bond formation. The synthesized compounds were evaluated for in vitro antibacterial activity against Bacillus subtilis, Escherichia coli, Staphylococcus aureus, Klebsiella pneumoniae, and anticancer properties against human breast cancer cell lines. The result showed that the zone of inhibition of C1 complex against S. aureus was 16.9 mm, which was closer to the standard drug. On the other hand, C4 complex exhibited the highest cytotoxicity with the lowest IC50 values of 73.05 μg·mL−1 against human breast cancer cell lines. In addition, molecular docking studies of ligand and complexes with 3TOT, 5MVR, 6D9T, and 6HZQ receptor proteins were conducted. S. aureus shows three hydrogen bond interactions (ASP281, VAL283, ILE36) with a binding affinity of −9.7 kcal·mol−1 for C1 complex over other complexes.

Abstract For a graph Q = ( V , E ) {\mathbb{Q}}=\left({\mathbb{V}},{\mathbb{E}}) , the transformation graph are defined as graphs with vertex set being V ( Q ) ∪ E ( Q ) {\mathbb{V}}\left({\mathbb{Q}})\cup {\mathbb{E}}\left({\mathbb{Q}}) and edge set is described following certain conditions. In comparison with the structural descriptor of the original graph Q {\mathbb{Q}} , the topological descriptor of its transformation graphs displays distinct characteristics related to the structure. Thus, a compound’s transformation graph descriptors can be used to model a variety of structural features of the underlying molecular structure and initiate a structural analysis. In this work, the concept of transformation graphs is extended giving rise to a novel class of graphs, the ( r , s ) \left(r,s) -generalised transformation graphs, whose vertex set is union of r r copies of V ( Q ) {\mathbb{V}}\left({\mathbb{Q}}) , and s s copies of E ( Q ) {\mathbb{E}}\left({\mathbb{Q}}) , where r , s ∈ N r,s\in N , and the edge set are defined under certain conditions. Furthermore, this class of graphs is analysed with the help of first Zagreb index. Mainly, there are eight transformation graphs based on the criteria for edge set, but under the concept of ( r , s ) \left(r,s) -generalised transformation graphs, infinite number of graphs can be described and analysed.

Abstract A novel zinc coordination polymer, [Zn3(pdc)2(bib)2.5(H2O)2] n ·4nH2O·0.5nbib (H3pdc = 3,5-pyrazoledicarboxylic acid, bib =1,4-bis(imidazol-1-ylmethyl)butane), has been synthesized through hydrothermal methods and characterized structurally by single-crystal X-ray diffraction. The polymer crystallizes in a triclinic space group P 1 ¯ P\bar{1} , with unit cell parameters of a = 12.5612, b = 12.8928, c = 16.5488 Å, α = 105.556°, β = 102.193°, γ = 93.817°, and Z = 2. Zinc(ii) ions possess trigonal bipyramidal coordination, connecting different pdc3− anions to form tri-nuclear subunits. These subunits are linked to 2D networks via bib ligands. Additionally, the fluorescence properties of 1 have been examined.

Abstract This study examines many innovative topological numbers and establishes mathematical interpretations for boron clusters and borophene coverings. The general Randic index, arithmetic index, and Albertson index are discussed in this work for the alpha sheets 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 vapour deposition and molecular beam epitaxy. Degree-based topological numbers 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 of a node’s degree in a molecular network, which indicates the number of neighbouring atoms that are directly connected to that node.

Abstract Reactions of AlEt 3 , Al( n -Oct) 3 , and AlEt 2 ( n -Oct) with thiophene-2-carbonyl chloride (TPCC) at Al/TPCC (molar ratio) <1 were respectively studied. The reaction produced ketone (ethyl thienyl ketone and/or n -octyl thienyl ketone) as the main product. The ketone yield reached the maximum after a very short reaction time and then slightly decreased. When TPCC solution was injected into AlEt 3 solution, faster addition of TPCC led to higher ketone yield, and higher temperature caused lower ketone yield. Increasing the size of R in AlR 3 from ethyl to n -octyl caused a marked decrease in ketone yield. The yield of the ketone produced from Al−Oct was about 1/5 of the ketone from Al-Et in the TPCC-AlEt 2 Oct reaction. The reaction system showed rapid color changes with time in the first few seconds. Based on the kinetic feature and reaction phenomena, a mechanistic model is proposed, in which the formation of [R′CO] + [AlR 3 Cl] − (R′ = thienyl) ion pair is much faster than that of R′COCl·AlR 3 donor–acceptor complex, and only the former is able to produce ketone. Though the formation of the R′COCl·AlR 3 complex drags behind that of the ion pair, ketone formation is completely depressed when all AlR 3 molecules are coordinated by acyl chloride or ketone.

Abstract Camptothecin is a naturally occurring alkaloid known for its significant and selective inhibition of the topoisomerase nuclear enzyme, a critical target in cancer treatment. IT-101, a polymeric drug conjugate of camptothecin, exhibits potent antiglioma activity in vitro , making it a highly promising candidate for cancer therapy. This conjugate not only slows the progression of various malignancies but also enhances the therapeutic efficacy of camptothecin. Topological indices, which are numerical values associated with the molecular structure of chemical compounds, serve as powerful tools for predicting physical properties and biological activities. Calculating these indices offers an efficient alternative to time-consuming and costly laboratory experiments. In this article, we first computed the M-polynomial and NM-polynomial of the camptothecin–polymer conjugate IT-101 structure. By applying various integration and differentiation formulas, we have computed different degree-based topological indices (TIs) for the camptothecin–polymer conjugate IT-101 structure. From an application perspective, we employed a linear regression model to estimate the physicochemical properties of 29 anticancer drugs using eccentricity-based TIs. The results demonstrate that two properties, namely, molecular weight and complexity, can be predicted with high accuracy using the first Zagreb eccentric index. The results may be useful to obtain insights into its molecular characteristics and potential applications in cancer treatment.

Abstract Two new coordination polymers, {(NH2(CH3)2)2[Zn2(bipy)(SO4)3]} n (1) and [Ni(phen)(SO4)(H2O)2] n (2) (bipy = 4,4′-bipyridine, phen = 1,10-phenanthroline) have been synthesized by using metal sulfate, nitrogen-containing ligands, and different template agents under solvothermal conditions and structurally characterized by single-crystal X-ray diffraction. 1 exhibits binuclear units and SO4 2− connect two Zn(ii) ions’ centers. It shows 2D layer structure and further extends into 3D supramolecular framework by N–H⋯S hydrogen bonds. Moreover, 2 possesses a 1D double-chain structure and form 2D layer by hydrogen-bonding interactions. And we further explored the infrared (IR) of 1 and 2 and luminescent properties of 1.