Nanofluids, which are suspensions of nanoparticles in base fluids, have demonstrated considerable potential in enhancing thermal conductivity, energy storage, and lubrication properties, as well as improving the cooling efficiency of electronic devices. Despite their promising applications, the industrial utilization of nanofluids remains in the early stages, with further research needed to fully explore their capabilities. This study investigates a generalized nanofluid model, incorporating fractal-fractional derivative (FFD), to better understand the thermophysical behaviors in vertical channel flow. The nanofluid consists of polystyrene nanoparticles uniformly dispersed in kerosene oil. An exact solution to the model is obtained by employing the Laplace transform technique (LTT) in combination with the numerical Zakian’s algorithm. The FFD operator with an exponential kernel is applied to extend the classical nanofluid model. Discretization of the generalized model is achieved using the Crank-Nicolson method, and numerical simulations are performed to solve the resulting equations. The study reveals that, at a nanoparticle volume fraction of 4% (0.04), the heat transfer rate of the nanofluid is significantly higher than that of the base fluid. Furthermore, the enhanced heat transfer leads to improvements in various thermophysical properties, such as viscosity, thermal expansion, and heat capacity, which are crucial for industrial applications. The numerical results are presented graphically to highlight the dependence of the flow and thermal dispersion characteristics on key physical factors. These findings suggest that the use of fractal-fractional models can provide a more accurate representation of nanofluid behavior, particularly for high-precision applications in heat transfer and energy systems.

Graph structures (GSs) have appeared as a robust mathematical framework for modelling and resolving complex combinatorial problems across diverse realms. At the same time, the linear Diophantine fuzzy set (LDFS) is a noteworthy expansion of the conventional concepts of the fuzzy set (FS), intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PFS), and q-Rung orthopair fuzzy set (q-ROFS). The LDFS framework introduces a flexible parameterization strategy that independently relaxes membership and non-membership restraints through reference parameters, thereby attaining enhanced expressiveness in apprehending ambiguous real-world phenomena. In this paper, a novel concept of linear Diophantine fuzzy graph structure (LDFGS) is introduced as a generalization of intuitionistic fuzzy graph structure (IFGS) and linear Diophantine fuzzy graph (LDFG) to GSs. Several cardinal fundamental notions in LDFGSs, including $\breve{\rho}_i$-edge, $\breve{\rho}_i$-path, strength of $\breve{\rho}_i$-path, $\breve{\rho}_i$-strength of connectedness, $\breve{\rho}_i$-degree of a vertex, degree of a vertex, total $\breve{\rho}_i$-degree of a vertex, and the total degree of a vertex in an LDFGS are discussed. Additionally, $\breve{\rho}_i$-size of an LDFGS, the size of an LDFGS, and the order of an LDFGS are studied. Meanwhile, the ideas of the maximal product of two LDFGSs, strong LDFGS, degree, and $\breve{\rho}_i$-degree of the maximal product are introduced with several concrete illustrations. To empirically validate the efficacy and practical utility of the proposed LDFGS framework, this study presents a case study analyzing road crime patterns across heterogeneous urban regions in Sindh province, Pakistan.

The forecasting of wheat commodity prices plays a crucial role in mitigating financial risks for stakeholders across the agricultural supply chain. In this study, the predictive performance of three models—Simple Moving Average (SMA), Extreme Gradient Boosting (XGBoost), and a hybrid SMA-XGBoost model—was evaluated to determine their efficacy in capturing both linear trends and complex nonlinear patterns inherent in wheat price data. A 10-lag structure was employed to integrate historical dependencies and seasonal fluctuations, thereby enhancing the accuracy of trend identification. The dataset was partitioned into training (75%) and testing (25%) subsets to facilitate an objective performance assessment. The XGBoost model, known for its capability in modelling nonlinear dependencies, demonstrated the highest forecasting precision, achieving a Mean Absolute Percentage Error (MAPE) of 1.64%. The hybrid SMA-XGBoost model, which leveraged the complementary strengths of both SMA and XGBoost, yielded a MAPE of 1.75%, outperforming the standalone SMA model, which exhibited a MAPE of 2.60%. While the hybrid model displayed slightly lower accuracy than XGBoost, it offered greater stability and robustness by effectively balancing trend extraction and nonlinear adaptability. These findings highlight the hybrid approach as a viable alternative to purely machine learning-based forecasting methods, particularly in scenarios requiring resilience to diverse market fluctuations. The proposed methodology provides a valuable tool for policymakers, agricultural producers, and market analysts seeking to enhance decision-making strategies and optimize risk management within the agricultural sector.