It was demonstrated by Bibi and Edjvet in ^[1] that any equation with a length of at most seven over torsion-free group can be solvable. This corroborates Levin's ^[2] assertion that any equation over a torsion-free group is solvable. It is demonstrated in this article that a singular equation of length eight over torsion-free groups is solvable.

Topological indices are mathematical values based on graph models of molecular structures that characterize significant properties in terms of chemical composition, reactivity, and physicochemical properties. In this paper, we are devoted to eccentricity-based indices of power graphs over finite groups and investigate their application in the context of molecular graphs. We calculated the Zagreb eccentricity indices, eccentric connectivity index, connective eccentricity index, (adjacent) eccentric distance sum index, and the Zagreb irregularity indices. In addition, we computed the Hosoya index for the mentioned graphs, which was one of the challenging aspects of this work. These findings enhance the theoretical foundation of graph-based indices and contribute to the quantitative description of molecular graphs.

The numerical approximation of stochastic partial differential equations (SPDEs), particularly those including q-diffusion, poses considerable challenges due to the requirements for high-order precision, stability amongst random perturbations, and processing efficiency. Because of their simplicity, conventional numerical techniques like the Euler-Maruyama method are frequently employed to solve stochastic differential equations; nonetheless, they may have low-order accuracy and lower stability in stiff or high-resolution situations. This study proposes a novel computational scheme for solving SPDEs arising from a stochastic SEIR model with q-diffusion and a general incidence rate function. A proposed computational scheme can be used to solve stochastic partial differential equations. For spatial discretization, a compact scheme is chosen. The compact scheme can provide a sixth-order accurate solution. The proposed scheme can be considered an extension of the Euler Maruyama method. Stability and consistency in the mean square sense are also provided. For application purposes, the stochastic SEIR model is considered using q-diffusion effects. The scheme is used to solve the stochastic model and compared with the Euler-Maruyama method. The scheme is also compared with nonstandard finite difference method for solving deterministic models. In both cases, it performs better than existing schemes. Incorporating q-diffusion further enhanced the model’s ability to represent realistic spatial-temporal disease dynamics, especially in scenarios where classical diffusion is insufficient.