Graphyne and Graphdiyne Nanoribbons reveal significant prospective with diverse applications. In electronics, they propose unique electronic properties for high-performance nanoscale devices, while in catalysis, their excellent surface area and reactivity sort them valuable catalyst supports for numerous chemical reactions, contributing to progresses in sustainable energy and environmental remediation. The topological indices (TIs) are numerical invariants that provide important information about the molecular topology of a given molecular graph. These indices are essential in QSAR/QSPR analysis and play a significant role in predicting various physico-chemical characteristics. In this article, we present a formula for computing reduced reverse (RR) degree-based topological indices for graphyne and graphdiyne nanoribbons, including the RR Zagreb indices, RR hyper-Zagreb indices, RR forgotten index, RR atom bond connectivity index, and RR Geometric-arithmetic index. We also execute a graph-theoretical analysis and comparison to demonstrate the critical significance and validate the acquired results. Our findings provide insights into the structural and chemical properties of these nanoribbons and contribute to the development of new materials for various applications.

This paper is concerned with the novel exact solitons for the truncated M-fractional (1+1)-dimensional nonlinear generalized Bretherton model with arbitrary constants. This model is used to explain the resonant nonlinear interaction between the waves in different phenomena, including fluid dynamics, plasma physics, ocean waves, and many others. A series of exact solitons, including bright, dark, periodic, singular, singular–bright, singular–dark, and other solitons are obtained by applying the extended sinh-Gordon equation expansion (EShGEE) and the modified $(G^{\prime }/G^2)$-expansion techniques. A novel definition of fractional derivative provides solutions that are distinct from previous solutions. Mathematica software was used to obtain and verify the solutions. The solutions are shown through 2D, 3D, and density plots. A stability process was conducted to verify that the solutions are exact and accurate. Modulation instability was used to determine the steady-state results for the corresponding equation.

Neurodegenerative diseases (NDDs) have received considerable interest from scientists for a long time due to their multifactorial nature. Alzheimer’s disease (AD) is of particular importance among pathologies, and despite approved drugs for its treatment, there is no effective pharmacological therapy to stop, halt, or repair neurodegeneration. The U.S. Food and Drug Administration (FDA) has approved certain medications to treat AD, including galantamine, donepezil, rivastigmine, memantine, tacrine, suvorexant, brexpiprazole, butein, and Licochalcone A. Topological indices and quantitative structure‐property relationships (QSPRs) are indispensable in drug discovery. They allow researchers to analyze, compare, and predict the properties of chemical compounds, thereby expediting the identification of promising drug candidates while minimizing experimental costs and efforts. Regression models are vital in QSPR analysis, especially when dealing with topological indices. They facilitate quantifying the relationship between chemical structure and properties, thereby facilitating drug design, material discovery, and other chemistry‐related applications. The objective of this study is to examine the efficacy of descriptors in correlating the physicochemical features of molecules associated with Alzheimer’s disease. We will use the computational method to calculate degree‐related, distance‐related, and eccentricity‐based topological indices for any chemical graph. QSPR models are developed utilizing degree‐based, distance‐based, and eccentricity‐based topological indices to estimate some physicochemical properties of AD drugs, including molecular weight (MW), boiling point (BP), topological polar surface area (TPSA), complexity (C), polarizability (P), and refractive index (R). The QSPRS studies are obtained using the linear regression technique. The present study found that the topological indices Randic index R, first Zagreb index M₁, and atom‐bond connectivity index ABC provide valuable insights into the structure‐activity relationships of different drugs and help in designing more effective combinations for treating Alzheimer’s disease.

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right $\hslash$-convexity ($LR$-$\hslash$-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated $\hslash$-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes.